probability and statistics binomial experiment definition

Binomial Experiments: An Explanation + Examples

Understanding binomial experiments is the first step to understanding the binomial distribution .

This tutorial defines a binomial experiment and provides several examples of experiments that are and are not considered to be binomial experiments.

Binomial Experiment: Definition

A binomial experiment is an experiment that has the following four properties:

1. The experiment consists of n repeated trials. The number n can be any amount. For example, if we flip a coin 100 times, then n = 100.

2. Each trial has only two possible outcomes. We often call outcomes either a “success” or a “failure” but a “success” is just a label for something we’re counting. For example, when we flip a coin we might call a head a “success” and a tail a “failure.”

3. The probability of success, denoted p , is the same for each trial. In order for an experiment to be a true binomial experiment, the probability of “success” must be the same for each trial. For example, when we flip a coin, the probability of getting heads (“success”) is always the same each time we flip the coin.

4. Each trial is independent . This simply means that the outcome of one trial does not affect the outcome of another trial. For example, suppose we flip a coin and it lands on heads. The fact that it landed on heads doesn’t change the probability that it will land on heads on the next flip. Each flip (i.e. each “trial”) is independent.

Examples of Binomial Experiments

The following experiments are all examples of binomial experiments.

Flip a coin 10 times. Record the number of times that it lands on tails.

This is a binomial experiment because it has the following four properties:

The experiment consists of n repeated trials. In this case, there are 10 trials.
Each trial has only two possible outcomes. The coin can only land on heads or tails.
The probability of success is the same for each trial . If we define “success” as landing on heads, then the probability of success is exactly 0.5 for each trial.
Each trial is independent . The outcome of one coin flip does not affect the outcome of any other coin flip.

Roll a fair 6-sided die 20 times. Record the number of times that a 2 comes up.

The experiment consists of n repeated trials. In this case, there are 20 trials.
Each trial has only two possible outcomes. If we define a 2 as a “success” then each time the die either lands on a 2 (a success) or some other number (a failure).
The probability of success is the same for each trial . For each trial, the probability that the die lands on a 2 is 1/6. This probability does not change from one trial to the next.
Each trial is independent . The outcome of one die roll does not affect the outcome of any other die roll.

Tyler makes 70% of his free-throw attempts. Suppose he makes 15 attempts. Record the number of baskets he makes.

The experiment consists of n repeated trials. In this case, there are 15 trials.
Each trial has only two possible outcomes. For each attempt, Tyler either makes the basket or misses it.
The probability of success is the same for each trial . For each trial, the probability that Tyler makes the basket is 70%. This probability does not change from one trial to the next.
Each trial is independent . The outcome of one free-throw attempt does not affect the outcome of any other free-throw attempt.

Examples that are not Binomial Experiments

Ask 100 people how old they are .

This is not a binomial experiment because there are more than two possible outcomes.

Roll a fair 6-sided die until a 5 comes up.

This is not a binomial experiment because there is not a pre-defined n number of trials. We have no idea how many rolls it will take until a 5 comes up.

Pull 5 cards from a deck of cards.

This is not a binomial experiment because the outcome of one trial (e.g. pulling a certain card from the deck) affects the outcome of future trials.

A Binomial Experiment Example & Solution

The following example shows how to solve a question about a binomial experiment.

You flip a coin 10 times. What is the probability that the coin lands on heads exactly 7 times?

Whenever we’re interested in finding the probability of n successes in a binomial experiment, we must use the following formula:

P(exactly k successes) = n C k * p k * (1-p) n-k

n: the number of trials
k: the number of successes
C: the symbol for “combination”
p: probability of success on a given trial

Plugging these numbers into the formula, we get:

P(7 heads) = 10 C 7 * 0.5 7 * (1-0.5) 10-7 = (120) * (.0078125) * (.125) = 0.11719 .

Thus, the probability that the coin lands on heads 7 times is 0.11719 .

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4.3 Binomial Distribution

There are three characteristics of a binomial experiment.

There are a fixed number of trials. Think of trials as repetitions of an experiment. The letter n denotes the number of trials.
There are only two possible outcomes, called "success" and "failure," for each trial. The letter p denotes the probability of a success on one trial, and q denotes the probability of a failure on one trial. p + q = 1.
The n trials are independent and are repeated using identical conditions. Because the n trials are independent, the outcome of one trial does not help in predicting the outcome of another trial. Another way of saying this is that for each individual trial, the probability, p , of a success and probability, q , of a failure remain the same. For example, randomly guessing at a true-false statistics question has only two outcomes. If a success is guessing correctly, then a failure is guessing incorrectly. Suppose Joe always guesses correctly on any statistics true-false question with probability p = 0.6. Then, q = 0.4. This means that for every true-false statistics question Joe answers, his probability of success ( p = 0.6) and his probability of failure ( q = 0.4) remain the same.

The outcomes of a binomial experiment fit a binomial probability distribution . The random variable X = the number of successes obtained in the n independent trials.

The mean, μ , and variance, σ 2 , for the binomial probability distribution are μ = np and σ 2 = npq . The standard deviation, σ , is then σ = n p q n p q .

Any experiment that has characteristics two and three and where n = 1 is called a Bernoulli Trial (named after Jacob Bernoulli who, in the late 1600s, studied them extensively). A binomial experiment takes place when the number of successes is counted in one or more Bernoulli Trials.

Example 4.9

At ABC College, the withdrawal rate from an elementary physics course is 30% for any given term. This implies that, for any given term, 70% of the students stay in the class for the entire term. A "success" could be defined as an individual who withdrew. The random variable X = the number of students who withdraw from the randomly selected elementary physics class.

The state health board is concerned about the amount of fruit available in school lunches. Forty-eight percent of schools in the state offer fruit in their lunches every day. This implies that 52% do not. What would a "success" be in this case?

Example 4.10

Suppose you play a game that you can only either win or lose. The probability that you win any game is 55%, and the probability that you lose is 45%. Each game you play is independent. If you play the game 20 times, write the function that describes the probability that you win 15 of the 20 times. Here, if you define X as the number of wins, then X takes on the values 0, 1, 2, 3, ..., 20. The probability of a success is p = 0.55. The probability of a failure is q = 0.45. The number of trials is n = 20. The probability question can be stated mathematically as P ( x = 15).

Try It 4.10

A trainer is teaching a rescued dolphin to catch live fish before returning it to the wild. The probability that the dolphin successfully catches a fish is 35%, and the probability that the dolphin does not successfully catch the fish is 65%. Out of 20 attempts, you want to find the probability that the dolphin succeeds 12 times. State the probability question mathematically.

Example 4.11

A coin has been altered to weight the outcome from 0.5 to 0.25 and flipped 5 times. Each flip is independent. What is the probability of getting more than 3 heads? Let X = the number of heads in 5 flips of the fair coin. X takes on the values 0, 1, 2, 3, 4, 5. Since the coin is altered to result in p = 0.25, q is 0.75. The number of trials is n = 5. State the probability question mathematically.

First develop fully the probability density function and graph the probability density function. With the fully developed probability density function we can simply read the solution to the question P x > 3 P x > 3 heads. P x > 3 = P x = 4 + P x = 5 = 0 . 0146 + 0 . 0007 = 0 . 0153 . P x > 3 = P x = 4 + P x = 5 = 0 . 0146 + 0 . 0007 = 0 . 0153 . We have added the two individual probabilities because of the addition rule from Probability Topics .

Figure 4.2 also allows us to see the link between the probability density function and probability and area. We also see in Figure 4.2 the skew of the binomial distribution when p is not equal to 0.5. In Figure 4.2 the distribution is skewed right as a result of μ = n p = 1 . 25 μ = n p = 1 . 25 because p = 0 . 25 p = 0 . 25 .

Try It 4.11

A fair, six-sided die is rolled ten times. Each roll is independent. You want to find the probability of rolling a one more than three times. State the probability question mathematically.

Example 4.12

Approximately 70% of statistics students do their homework in time for it to be collected and graded. Each student does homework independently. In a statistics class of 50 students, what is the probability that at least 40 will do their homework on time? Students are selected randomly.

a. This is a binomial problem because there is only a success or a __________, there are a fixed number of trials, and the probability of a success is 0.70 for each trial.

b. If we are interested in the number of students who do their homework on time, then how do we define X ?

c. What values does x take on?

d. What is a "failure," in words?

e. If p + q = 1, then what is q ?

f. The words "at least" translate as what kind of inequality for the probability question P ( x ____ 40).

b. X = the number of statistics students who do their homework on time

c. 0, 1, 2, …, 50

d. Failure is defined as a student who does not complete their homework on time.

The probability of a success is p = 0.70. The number of trials is n = 50.

e. q = 0.30

f. greater than or equal to (≥) The probability question is P ( x ≥ 40).

Try It 4.12

Sixty-five percent of people pass the state driver’s exam on the first try. A group of 50 individuals who have taken the driver’s exam is randomly selected. Give two reasons why this is a binomial problem.

Notation for the Binomial: B = Binomial Probability Distribution Function

X ~ B ( n , p )

Read this as " X is a random variable with a binomial distribution." The parameters are n and p ; n = number of trials, p = probability of a success on each trial.

Example 4.13

It has been stated that about 41% of adult workers have a high school diploma but do not pursue any further education. If 20 adult workers are randomly selected, find the probability that at most 12 of them have a high school diploma but do not pursue any further education. How many adult workers do you expect to have a high school diploma but do not pursue any further education?

Let X = the number of workers who have a high school diploma but do not pursue any further education.

X takes on the values 0, 1, 2, ..., 20 where n = 20, p = 0.41, and q = 1 – 0.41 = 0.59. X ~ B (20, 0.41)

Find P ( x ≤ 12). P ( x ≤ 12) = 0.9738. (calculator or computer)

Using the TI-83, 83+, 84, 84+ Calculator

Go into 2 nd DISTR. The syntax for the instructions are as follows:

To calculate ( x = value): binompdf( n , p , number) if "number" is left out, the result is the binomial probability table. To calculate P ( x ≤ value): binomcdf( n , p , number) if "number" is left out, the result is the cumulative binomial probability table. For this problem: After you are in 2 nd DISTR , arrow down to binomcdf . Press ENTER . Enter 20,0.41,12). The result is P ( x ≤ 12) = 0.9738.

If you want to find P ( x = 12), use the pdf (binompdf). If you want to find P ( x > 12), use 1 - binomcdf(20,0.41,12).

The probability that at most 12 workers have a high school diploma but do not pursue any further education is 0.9738.

The graph of X ~ B (20, 0.41) is as follows:

The y -axis contains the probability of x , where X = the number of workers who have only a high school diploma.

The number of adult workers that you expect to have a high school diploma but not pursue any further education is the mean, μ = np = (20)(0.41) = 8.2.

The formula for the variance is σ 2 = npq . The standard deviation is σ = n p q n p q . σ = ( 20 ) ( 0.41 ) ( 0.59 ) ( 20 ) ( 0.41 ) ( 0.59 ) = 2.20.

Try It 4.13

About 32% of students participate in a community volunteer program outside of school. If 30 students are selected at random, find the probability that at most 14 of them participate in a community volunteer program outside of school. Use the TI-83+ or TI-84 calculator to find the answer.

Example 4.14

In the 2013 Jerry’s Artarama art supplies catalog, there are 560 pages. Eight of the pages feature signature artists. Suppose we randomly sample 100 pages. Let X = the number of pages that feature signature artists.

What values does x take on?
the probability that two pages feature signature artists
the probability that at most six pages feature signature artists
the probability that more than three pages feature signature artists.
Using the formulas, calculate the (i) mean and (ii) standard deviation.
x = 0, 1, 2, 3, 4, 5, 6, 7, 8
P ( x = 2) = binompdf ( 100 , 8 560 , 2 ) ( 100 , 8 560 , 2 ) = 0.2466
P ( x ≤ 6) = binomcdf ( 100 , 8 560 , 6 ) ( 100 , 8 560 , 6 ) = 0.9994
P ( x > 3) = 1 – P ( x ≤ 3) = 1 – binomcdf ( 100 , 8 560 , 3 ) ( 100 , 8 560 , 3 ) = 1 – 0.9443 = 0.0557
Mean = np = (100) ( 8 560 ) ( 8 560 ) = 800 560 800 560 ≈ 1.4286
Standard Deviation = n p q n p q = ( 100 ) ( 8 560 ) ( 552 560 ) ( 100 ) ( 8 560 ) ( 552 560 ) ≈ 1.1867

Try It 4.14

According to a Gallup poll, 60% of American adults prefer saving over spending. Let X = the number of American adults out of a random sample of 50 who prefer saving to spending.

What is the probability distribution for X ?
the probability that 25 adults in the sample prefer saving over spending
the probability that at most 20 adults prefer saving
the probability that more than 30 adults prefer saving
Using the formulas, calculate the (i) mean and (ii) standard deviation of X .

Example 4.15

The lifetime risk of developing cancer is about one in 67 (1.5%). Suppose we randomly sample 200 people. Let X = the number of people who will develop cancer.

Use your calculator to find the probability that at most eight people develop cancer
Is it more likely that five or six people will develop cancer? Justify your answer numerically.
X ~ B 200 , 0 . 015 X ~ B 200 , 0 . 015
Mean = n p = 200 0 . 015 = 3 Mean = n p = 200 0 . 015 = 3 Standard Deviation = n p q = 200 ( 0 . 015 ) ( 0 . 985 ) = 1 . 719 Standard Deviation = n p q = 200 ( 0 . 015 ) ( 0 . 985 ) = 1 . 719
P x ≤ 8 = 0 . 9965 P x ≤ 8 = 0 . 9965
The probability that five people develop cancer is 0.1011. The probability that six people develop cancer is 0.0500.

Try It 4.15

During a certain NBA season, a player for the Los Angeles Clippers had the highest field goal completion rate in the league. This player scored with 61.3% of his shots. Suppose you choose a random sample of 80 shots made by this player during the season. Let X = the number of shots that scored points.

Use your calculator to find the probability that this player scored with 60 of these shots.
Find the probability that this player scored with more than 50 of these shots.

Example 4.16

The following example illustrates a problem that is not binomial. It violates the condition of independence. ABC College has a student advisory committee made up of ten staff members and six students. The committee wishes to choose a chairperson and a recorder. What is the probability that the chairperson and recorder are both students? The names of all committee members are put into a box, and two names are drawn without replacement . The first name drawn determines the chairperson and the second name the recorder. There are two trials. However, the trials are not independent because the outcome of the first trial affects the outcome of the second trial. The probability of a student on the first draw is 6 16 6 16 . The probability of a student on the second draw is 5 15 5 15 , when the first draw selects a student. The probability is 6 15 6 15 , when the first draw selects a staff member. The probability of drawing a student's name changes for each of the trials and, therefore, violates the condition of independence.

Try It 4.16

A lacrosse team is selecting a captain. The names of all the seniors are put into a hat, and the first three that are drawn will be the captains. The names are not replaced once they are drawn (one person cannot be two captains). You want to see if the captains all play the same position. State whether this is binomial or not and state why.

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Binomial Distribution

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The binomial distribution is, in essence, the probability distribution of the number of heads resulting from flipping a weighted coin multiple times. It is useful for analyzing the results of repeated independent trials, especially the probability of meeting a particular threshold given a specific error rate, and thus has applications to risk management . For this reason, the binomial distribution is also important in determining statistical significance .

Formal Definition

Finding the binomial distribution, properties of the binomial distribution, practical applications, binomial test.

A Bernoulli trial , or Bernoulli experiment , is an experiment satisfying two key properties:

There are exactly two complementary outcomes, success and failure.
The probability of success is the same every time the experiment is repeated.

A binomial experiment is a series of $n$ Bernoulli trials, whose outcomes are independent of each other. A random variable , $X$, is defined as the number of successes in a binomial experiment. Finally, a binomial distribution is the probability distribution of $X$.

For example, consider a fair coin. Flipping the coin once is a Bernoulli trial, since there are exactly two complementary outcomes (flipping a head and flipping a tail), and they are both $\frac{1}{2}$ no matter how many times the coin is flipped. Note that the fact that the coin is fair is not necessary; flipping a weighted coin is still a Bernoulli trial.

A binomial experiment might consist of flipping the coin 100 times, with the resulting number of heads being represented by the random variable $X$. The binomial distribution of this experiment is the probability distribution of $X.$

Determining the binomial distribution is straightforward but computationally tedious. If there are $n$ Bernoulli trials, and each trial has a probability $p$ of success, then the probability of exactly $k$ successes is

\[\binom{n}{k}p^k(1-p)^{n-k}.\]

This is written as $\text{Pr}(X=k)$, denoting the probability that the random variable $X$ is equal to $k$, or as $b(k;n,p)$, denoting the binomial distribution with parameters $n$ and $p$.

The above formula is derived from choosing exactly $k$ of the $n$ trials to result in successes, for which there are $\binom{n}{k}$ choices, then accounting for the fact that each of the trials marked for success has a probability $p$ of resulting in success, and each of the trials marked for failure has a probability $1-p$ of resulting in failure. The binomial coefficient $\binom{n}{k}$ lends its name to the binomial distribution.

Consider a weighted coin that flips heads with probability $0.25$. If the coin is flipped 5 times, what is the resulting binomial distribution? This binomial experiment consists of 5 trials, a $p$-value of $0.25$, and the number of successes is either 0, 1, 2, 3, 4, or 5. Therefore, the above formula applies directly: \[\begin{align} \text{Pr}(X=0) &= b(0;5,0.25) = \binom{5}{0}(0.25)^0(0.75)^5 \approx 0.237\\ \text{Pr}(X=1) &= b(1;5,0.25) = \binom{5}{1}(0.25)^1(0.75)^4 \approx 0.396\\ \text{Pr}(X=2) &= b(2;5,0.25) = \binom{5}{2}(0.25)^2(0.75)^3 \approx 0.263\\ \text{Pr}(X=3) &= b(3;5,0.25) = \binom{5}{3}(0.25)^3(0.75)^2 \approx 0.088\\ \text{Pr}(X=4) &= b(4;5,0.25) = \binom{5}{4}(0.25)^4(0.75)^1 \approx 0.015\\ \text{Pr}(X=5) &= b(5;5,0.25) = \binom{5}{5}(0.25)^5(0.75)^0 \approx 0.001. \end{align}\] It's worth noting that the most likely result is to flip one head, which is explored further below when discussing the mode of the distribution. $_\square$

This can be represented pictorially, as in the following table:

The binomial distribution $b(5,0.25)$

You have an (extremely) biased coin that shows heads with probability 99% and tails with probability 1%. To test the coin, you tossed it 100 times.

What is the approximate probability that heads showed up exactly $ 99 $ times?

A fair coin is flipped 10 times. What is the probability that it lands on heads the same number of times that it lands on tails?

Give your answer to three decimal places.

There are several important values that give information about a particular probability distribution. The most important are as follows:

The mean , or expected value , of a distribution gives useful information about what average one would expect from a large number of repeated trials.
The median of a distribution is another measure of central tendency, useful when the distribution contains outliers (i.e. particularly large/small values) that make the mean misleading.
The mode of a distribution is the value that has the highest probability of occurring.
The variance of a distribution measures how "spread out" the data is. Related is the standard deviation , the square root of the variance, useful due to being in the same units as the data.

Three of these values--the mean, mode, and variance--are generally calculable for a binomial distribution. The median, however, is not generally determined.

The mean of a binomial distribution is intuitive:

The mean of $b(n,p)$ is $np.$

In other words, if an unfair coin that flips heads with probability $p$ is flipped $n$ times, the expected result would be $np$ heads.

Let $X_1, X_2, \ldots, X_n$ be random variables representing the Bernoulli trial with probability $p$ of success. Then $X = X_1 + X_2 + \cdots + X_n$, by definition. By linearity of expectation , \[E[X]=E[X_1+X_2+\cdots+X_n]=E[X_1]+E[X_2]+\cdots+E[X_n]=\underbrace{p+p+\cdots+p}_{n\text{ times}}=np.\ _\square\]

You have an (extremely) biased coin that shows heads with 99% probability and tails with 1% probability.

If you toss it 100 times, what is the expected number of times heads will come up?

This problem is part of the set Extremely Biased Coins.

A similar strategy can be used to determine the variance:

The variance of $b(n,p)$ is $np(1-p)$.

Since variance is additive, a similar proof to the above can be used: \[ \begin{align*} \text{Var}[X] &= \text{Var}(X_1 + X_2 + \cdots + X_n) \\ &= \text{Var}(X_1) + \text{Var}(X_2) + \cdots + \text{Var}(X_n) \\ &= \underbrace{p(1-p)+p(1-p)+\cdots+p(1-p)}_{n\text{ times}} \\ &= np(1-p) \end{align*} \] since the variance of a single Bernoulli trial is $p(1-p)$. $_\square$

The mode, however, is slightly more complicated. In most cases the mode is $\lfloor (n+1)p \rfloor$, but if $(n+1)p$ is an integer, both $(n+1)p$ and $(n+1)p-1$ are modes. Additionally, in the trivial cases of $p=0$ and $p=1$, the modes are 0 and $n,$ respectively.

The mode of $b(n,p)$ is

\[ \text{mode} = \begin{cases} 0 & \text{if } p = 0 \\ n & \text{if } p = 1 \\ (n+1)\,p\ \text{ and }\ (n+1)\,p - 1 &\text{if }(n+1)p\in\mathbb{Z} \\ \big\lfloor (n+1)\,p\big\rfloor & \text{if }(n+1)p\text{ is 0 or a non-integer}. \end{cases} \]

Daniel has a weighted coin that flips heads $\frac{2}{5}$ of the time and tails $\frac{3}{5}$ of the time. If he flips it $9$ times, the probability that it will show heads exactly $n$ times is greater than or equal to the probability that it will show heads exactly $k$ times, for all $k=0, 1,\dots, 9, k\ne n$.

If the probability that the coin will show heads exactly $n$ times in $9$ flips is $\frac{p}{q}$ for positive coprime integers $p$ and $q$, then find the last three digits of $p$.

The binomial distribution is applicable to most situations in which a specific target result is known, by designating the target as "success" and anything other than the target as "failure." Here is an example:

A die is rolled 3 times. What is the probability that no sixes occur? In this binomial experiment, rolling anything other than a 6 is a success and rolling a 6 is failure. Since there are three trials, the desired probability is \[b\left(3;3,\frac{5}{6}\right)=\binom{3}{3}\left(\frac{5}{6}\right)^3\left(\frac{1}{6}\right)^0 \approx .579.\] This could also be done by designating rolling a 6 as a success, and rolling anything else as failure. Then the desired probability would be \[b\left(0;3,\frac{1}{6}\right)=\binom{3}{0}\left(\frac{1}{6}\right)^0\left(\frac{5}{6}\right)^3 \approx .579\] just as before. $_\square$

The binomial distribution is also useful in analyzing a range of potential results, rather than just the probability of a specific one:

A manufacturer of widgets knows that 20% of the widgets he produces are defective. If he produces 10 widgets per day, what is the probability that at most two of them are defective? In this binomial experiment, manufacturing a working widget is a success and manufacturing a defective widget is a failure. The manufacturer needs at least 8 successes, making the probability \[ \begin{align*} b(8;10,0.8)+b(9;10,0.8)+b(10;10,0.8) &=\binom{10}{8}(0.8)^8(0.2)^2+\binom{10}{9}(0.8)^9(0.2)^1+\binom{10}{10}(0.8)^{10} \\\\ &\approx 0.678. \ _\square \end{align*} \]

This example also illustrates an important clash with intuition: generally, one would expect that an 80% success rate is appropriate when requiring 8 of 10 widgets to not be defective. However, the above calculation shows that an 80% success rate only results in at least 8 successes less than 68% of the time!

This calculation is especially important for a related reason: since the manufacturer knows his error rate and his quota, he can use the binomial distribution to determine how many widgets he must produce in order to earn a sufficiently high probability of meeting his quota of non-defective widgets.

Related to the final note of the last section, the binomial test is a method of testing for statistical significance . Most commonly, it is used to reject the null hypothesis of uniformity; for example, it can be used to show that a coin or die is unfair. In other words, it is used to show that the given data is unlikely under the assumption of fairness, so that the assumption is likely false.

A coin is flipped 100 times, and the results are 61 heads and 39 tails. Is the coin fair? The null hypothesis is that the coin is fair, in which case the probability of flipping at least 61 heads is \[\sum_{i=61}^{100}b(i;100,0.5) = \sum_{i=61}^{100}\binom{100}{i}(0.5)^{100} \approx 0.0176,\] or $1.76\%$. Determining whether this result is statistically significant depends on the desired confidence level; this would be enough to reject the null hypothesis at the 5% level, but not the 1% one. As the most commonly used confidence level is the 5% one, this would generally be considered sufficient to conclude that the coin is unfair. $_\square$

Geometric Distribution
Poisson Distribution

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Binomial distribution

by Marco Taboga , PhD

The binomial distribution is a univariate discrete distribution used to model the number of favorable outcomes obtained in a repeated experiment.

Table of contents

How the distribution is used

What you need to know, relation to the bernoulli distribution, expected value, moment generating function, characteristic function, distribution function, solved exercises.

Consider an experiment having two possible outcomes: either success or failure.

Suppose that the experiment is repeated several times and the repetitions are independent of each other.

The distribution of the number of experiments in which the outcome turns out to be a success is called binomial distribution.

Chart of binomial distribution with interactive calculator

A binomial distribution can be seen as a sum of mutually independent Bernoulli random variables that take value 1 in case of success of the experiment and value 0 otherwise.

This connection between the binomial and Bernoulli distributions will be illustrated in detail in the remainder of this lecture and will be used to prove several properties of the binomial distribution.

Before proceeding, you are advised to study the lecture on the Bernoulli distribution .

The binomial distribution is characterized as follows.

The binomial distribution is intimately related to the Bernoulli distribution. The following propositions show how.

binocdf(x,n,p)

returns the value of the distribution function at the point x when the parameters of the distribution are n and p .

You can also use the calculator at the top of this page.

Below you can find some exercises with explained solutions.

How to cite

Please cite as:

Taboga, Marco (2021). "Binomial distribution", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/probability-distributions/binomial-distribution.

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Chapter 4: Discrete Random Variables

4.3 Binomial Distribution

Learning objectives.

By the end of this section, you should be able to:

Identify the components of a binomial experiment
Use the formulas for a binomial random variable to compute mean, variance, and standard deviation

Binomial Experiments

There are three characteristics of a binomial experiment.

There are a fixed number of trials. Think of trials as repetitions of an experiment. The letter [latex]n[/latex] denotes the number of trials.
There are only two possible outcomes, called “success” and “failure,” for each trial. The letter [latex]p[/latex] denotes the probability of a success on one trial, and [latex]q[/latex] denotes the probability of a failure on one trial [latex]p + q = 1[/latex] .
The [latex]n[/latex] trials are independent and are repeated using identical conditions. Because the [latex]n[/latex] trials are independent, the outcome of one trial does not help in predicting the outcome of another trial. Another way of saying this is that for each individual trial, the probability, [latex]p[/latex] , of a success and probability, [latex]q[/latex] , of a failure remain the same.

For example, randomly guessing at a true-false statistics question has only two outcomes. If a success is guessing correctly, then a failure is guessing incorrectly. Suppose Taylor consistently guesses correctly on any statistics true-false question with probability [latex]p = 0.6[/latex] . Then, [latex]q = 0.4.[/latex] Consistency in guessing means that for every true-false statistics question Taylor answers, the probability of success ([latex]p = 0.6[/latex] ) and the probability of failure ([latex]q = 0.4[/latex] ) remain the same, a necessary criteria for a situation to be binomial.

Any experiment that has characteristics two and three and where [latex]n = 1[/latex] is called a Bernoulli Trial (named after Jacob Bernoulli who, in the late 1600s, studied them extensively). A binomial experiment takes place when the number of successes is counted in one or more Bernoulli Trials.

The outcomes of a binomial experiment fit a binomial probability distribution . The random variable [latex]X =[/latex] the number of successes obtained in the [latex]n[/latex] independent trials.

Notation for the Binomial

We use [latex]B[/latex] to represent the binomial probability distribution, and when [latex]X[/latex] fits the binomial distribution, we write [latex]X \sim B(n,p)[/latex]. Read this as “[latex]X[/latex] is a random variable with a binomial distribution with parameters [latex]n[/latex] and [latex]p[/latex],” which again represent the number of trials and the probability of “success.”

At ABC College, the withdrawal rate from an elementary physics course is 30% for any given term. This implies that, for any given term, 70% of the students stay in the class for the entire term. A “success” could be defined as an individual who withdrew. The random variable [latex]X =[/latex] the number of students who withdraw from the randomly selected elementary physics class.

If at the start of a particular term, 300 students are enrolled in the elementary physics course, [latex]n = 300[/latex] and [latex]p = 0.3[/latex] and [latex]X \sim B(300, 0.3)[/latex]. The possible outcomes are [latex]x = 0, \ldots, 300[/latex] and the probability [latex]P(X=x)[/latex] is the probability that [latex]x[/latex] students will withdraw during the term.

a. This is a binomial problem because there is only a success or a [latex]\underline{\hspace{20pt}}[/latex], there are a fixed number of trials, and the probability of a success is 0.70 for each trial.

b. If we are interested in the number of students who do their homework on time, then how do we define [latex]X[/latex]?

c. What values does [latex]x[/latex] take on?

d. What is a “failure,” in words?

e. If [latex]p + q = 1[/latex], then what is [latex]q[/latex]?

f. The words “at least” translate as what kind of inequality for the probability question [latex]P(X \underline{\hspace{20pt}} 40)[/latex].

Experiments That Are Not Binomial

Here are some common experiments that are not binomial:

Flipping a coin until you get one head. While there are only two outcomes (heads and tails) to each coin flip, and while the probability of getting a head on each flip is consistent, the number of trials will vary.
Most characteristics about people, such as weight, height, ethnicity, and gender. When we survey a group of people and ask for their weight, we get far more than two responses. When surveying ethnicity, as discussed in Chapter 1 an Other or Unknown category is needed to include all people in our results, specifically people who did not feel they fit into any of the ethnicity categories or declined to respond. While gender has historically been treated as a binomial outcome (male and female), not all people will feel they fit those two categories, so we may lose information and reliability by modeling gender as binomial.
Drawing cards without replacement. While you can specify a set number of [latex]n[/latex] trials, and you can set it up as two outcomes (success = red, failure = black), the draws are not independent. If you draw a red on the first draw, it is now slightly more likely to draw black than red on the second draw.

ABC College has a student advisory committee made up of ten staff members and six students. The committee wishes to choose a chairperson and a recorder by putting the names of all committee members into a box, and two names are drawn without replacement. The first name drawn determines the chairperson and the second name the recorder. Suppose the college wishes to find the probability that the chairperson and recorder are both students. Is this binomial?

Suppose 55% of people pass the state driver’s exam on the first try. You survey fifty randomly chosen adults in the state.

Which of these are binomial problems?

The number of the adults who have a driver’s license.
The number of the adults who got their driver’s license on the first try.
The number of times each adult has taken the driver’s exam.

Binomial Probability Function

Once we have decided we can use the binomial for a given situation, we can use the binomial probability function to find the probability of a specific number of successes, [latex]P(X=x)[/latex]. The binomial PMF is made up of two parts:

First, we need to find out how many different ways we can get x successes in n trials. To do this we can use the “Choose” function, also called the binomial coefficient, written as:

nCx = [latex]=\binom nx =\frac{n!}{x!(n-x)!}[/latex]

Note: The ! mark is the factorial operator.

The next part gives us the probability of a single one of those ways to get x successes in n trials. We can do this by using our independent multiplication rule. We multiply the probability of success ([latex]p[/latex]) raised to the number of successes ([latex]x[/latex]) by the probability of failure ([latex]q=1-p[/latex]) raised to the number of failures ([latex]n-x[/latex]).

[latex]p^x q^{(n-x)}[/latex]

Since we know each of these ways are equally likely and how many ways are possible we can now put the two pieces together. We multiply the probability of one way by how many we have to give us our overall probability of x successes in n trials.

[latex]P(X = x) = \frac{n!}{x!(n-x)!} p^x q^{(n-x)}[/latex]

Unfortunately the binomial does not have a nice form of CDF, but it is simply the sum of PDFs up until that point. Consider the following example to demonstrate this point.

It has been stated that about 41% of adult workers have a high school diploma but do not pursue any further education. Twenty adult workers are randomly selected.

Let [latex]X =[/latex] the number of workers who have a high school diploma but do not pursue any further education.

Then [latex]X[/latex] takes on the values [latex]0, 1, 2, \ldots, 20[/latex] where [latex]n = 20, p = 0.41[/latex], and [latex]q = 1-0.41 = 0.59[/latex]. Finally, [latex]X \sim B(20,0.41).[/latex]

The y -axis contains the probability of [latex]x[/latex], where [latex]X =[/latex] the number of workers who have only a high school diploma.

The graph of [latex]X \sim B(20, 0.41)[/latex] is as follows:

Histogram of Binomial Distribution with 20 trials and success probability 0.41

Find the probability that:

(a) Exactly 12 of them have a high school diploma

(b) At most 12 of them have a high school diploma but do not pursue any further education. How many adult workers do you expect to have a high school diploma but do not pursue any further education?

To compute binomial probabilities on a graphing calculator, g o into 2 nd DISTR. The syntax for the instructions are as follows:

To calculate [latex]P(X = x)[/latex]: binompdf( n , p , x). If [latex]x[/latex] is left out, the result is the binomial probability table.

To calculate [latex]P(X \leq x)[/latex]: binomcdf( n , p , x). If [latex]x[/latex] is left out, the result is the cumulative binomial probability table.

For the example problem: After you are in 2 nd DISTR, arrow down to binomcdf(. Press ENTER. Enter 20,0.41,12). The result is [latex]P(X \leq 12) = 0.9738[/latex].

If you wanted to instead find [latex]P(X>12)[/latex], use 1 – binomcdf(20,0.41,12).

In Excel, both binomial probabilities are computed using BINOM.DIST(x, n, p, True/False), where False computes [latex]P(X=x)[/latex] and is equivalent to binompdf, and True computes [latex]P(X \leq x)[/latex] and is equivalent to binomcdf.

About 32% of students participate in a community volunteer program outside of school. If 30 students are selected at random, find:

(a) The probability that exactly 14 of them participate in a community volunteer program outside of school. First try plugging in to the binomial formula by hand, then check yourself with technology.

(b) The probability that exactly 14 of them participate in a community volunteer program outside of school. Rely on technology for this cumulative probability.

Expected Value and Standard Deviation

The mean, [latex]\mu[/latex] , and variance, [latex]\sigma^2[/latex] , for the binomial probability distribution are [latex]\mu = np[/latex] and [latex]\sigma^2 = npq[/latex]. The standard deviation is then [latex]\sigma = \sqrt{npq}[/latex].

What values does x take on?
the probability that two pages feature signature artists
the probability that at most six pages feature signature artists
the probability that more than three pages feature signature artists.
Using the formulas, calculate the (i) mean and (ii) standard deviation.
x = 0, 1, 2, 3, 4, 5, 6, 7, 8
P ( x = 2) = binompdf[latex]\left(100,\frac{8}{560},2\right)[/latex] = 0.2466
P ( x ≤ 6) = binomcdf[latex]\left(100,\frac{8}{560},6\right)[/latex] = 0.9994
P ( x > 3) = 1 – P ( x ≤ 3) = 1 – binomcdf[latex]\left(100,\frac{8}{560},3\right)[/latex] = 1 – 0.9443 = 0.0557
Mean = np = (100)[latex]\left(\frac{8}{560}\right)[/latex] = [latex]\frac{800}{560}[/latex] ≈ 1.4286
Standard Deviation = [latex]\sqrt{npq}[/latex] = [latex]\sqrt{\left(100\right)\left(\frac{8}{560}\right)\left(\frac{552}{560}\right)}[/latex] ≈ 1.1867

According to a Gallup poll, 60% of American adults prefer saving over spending. Let X = the number of American adults out of a random sample of 50 who prefer saving to spending.

What is the probability distribution for X ?
the probability that 25 adults in the sample prefer saving over spending
the probability that at most 20 adults prefer saving
the probability that more than 30 adults prefer saving
Using the formulas, calculate the (i) mean and (ii) standard deviation of X.

During the 2013 regular NBA season, DeAndre Jordan of the Los Angeles Clippers had the highest field goal completion rate in the league. DeAndre scored with 61.3% of his shots. Suppose you choose a random sample of 80 shots made by DeAndre during the 2013 season. Let X = the number of shots that scored points.

Using the formulas, calculate the (i) mean and (ii) standard deviation of X .
Use your calculator to find the probability that DeAndre scored with 60 of these shots.
Find the probability that DeAndre scored with more than 50 of these shots.

“Access to electricity (% of population),” The World Bank, 2013. Available online at http://data.worldbank.org/indicator/EG.ELC.ACCS.ZS?order=wbapi_data_value_2009%20wbapi_data_value%20wbapi_data_value-first&sort=asc (accessed May 15, 2015).

“Distance Education.” Wikipedia. Available online at http://en.wikipedia.org/wiki/Distance_education (accessed May 15, 2013).

“NBA Statistics – 2013,” ESPN NBA, 2013. Available online at http://espn.go.com/nba/statistics/_/seasontype/2 (accessed May 15, 2013).

Newport, Frank. “Americans Still Enjoy Saving Rather than Spending: Few demographic differences seen in these views other than by income,” GALLUP® Economy, 2013. Available online at http://www.gallup.com/poll/162368/americans-enjoy-saving-rather-spending.aspx (accessed May 15, 2013).

Pryor, John H., Linda DeAngelo, Laura Palucki Blake, Sylvia Hurtado, Serge Tran. The American Freshman: National Norms Fall 2011 . Los Angeles: Cooperative Institutional Research Program at the Higher Education Research Institute at UCLA, 2011. Also available online at http://heri.ucla.edu/PDFs/pubs/TFS/Norms/Monographs/TheAmericanFreshman2011.pdf (accessed May 15, 2013).

“The World FactBook,” Central Intelligence Agency. Available online at https://www.cia.gov/library/publications/the-world-factbook/geos/af.html (accessed May 15, 2013).

“What are the key statistics about pancreatic cancer?” American Cancer Society, 2013. Available online at http://www.cancer.org/cancer/pancreaticcancer/detailedguide/pancreatic-cancer-key-statistics (accessed May 15, 2013).

Media Attributions

an experiment with the following characteristics: 1. There are only two possible outcomes called “success” and “failure” for each trial. 2. The probability p of a success is the same for any trial (so the probability q = 1 − p of a failure is the same for any trial).

A statistical experiment that satisfies three conditions: 1. There are a fixed number of trials, n . 2. There are only two possible outcomes, called "success" and "failure," for each trial. The letter p denotes the probability of a success on one trial, and q denotes the probability of a failure on one trial. 3. The n trials are independent and are repeated using identical conditions.

A discrete random variable which arises from Bernoulli trials; there are a fixed number, n , of independent trials with two outcomes called success and failure with probability p and q respectively. The binomial random variable X is the number of successes in n trials, denoted [latex]X \sim B(n,p)[/latex]. The mean is [latex]\mu = np[\latex] and the standard deviation is [latex]\sigma = \sqrt{npq}[/latex]. The probability of exactly [latex]x[/latex] successes in n trials is [latex]P\left(X=x\right)=\left(\genfrac{}{}{0}{}{n}{x}\right){p}^{x}{q}^{n-x}[/latex].

A function that gives the probability that a discrete random variable (X) is exactly equal to some value (x)

A function that gives the probability that a random variable takes a value less than or equal to x

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Module 4: Discrete Random Variables

What are binomial experiments, learning outcomes.

Describe the three characteristics of a binomial experiment

There are three characteristics of a binomial experiment .

There are a fixed number of trials. Think of trials as repetitions of an experiment. The letter [latex]n[/latex] denotes the number of trials.
There are only two possible outcomes, called “success” and “failure,” for each trial. The letter [latex]p[/latex] denotes the probability of a success on one trial, and [latex]q[/latex] denotes the probability of a failure on one trial. [latex]p+q=1[/latex].
The [latex]n[/latex] trials are independent and are repeated using identical conditions. Because the [latex]n[/latex] trials are independent, the outcome of one trial does not help in predicting the outcome of another trial. Another way of saying this is that for each individual trial, the probability, [latex]p[/latex] of a success and probability, [latex]q[/latex], of a failure remain the same. For example, randomly guessing at a true-false statistics question has only two outcomes. If a success is guessing correctly, then a failure is guessing incorrectly. Suppose Joe always guesses correctly on any statistics true-false question with probability [latex]p=0.6[/latex]. Then, [latex]q=0.4[/latex]. This means that for every true-false statistics question Joe answers, his probability of success [latex](p=0.6)[/latex] and his probability of failure [latex](q=0.4)[/latex] remain the same.

The outcomes of a binomial experiment fit a binomial probability distribution . The random variable [latex]X=[/latex] the number of successes obtained in the [latex]n[/latex] independent trials.

The mean, [latex]\mu[/latex], and variance, [latex]\sigma^{2}[/latex], for the binomial probability distribution are [latex]\mu=np[/latex] and [latex]\sigma^{2}=npq[/latex]. The standard deviation, [latex]\sigma[/latex], is then [latex]\sigma=\sqrt{{{n}{p}{q}}}[/latex].

Any experiment that has characteristics two and three and where [latex]n=1[/latex] is called a Bernoulli Trial (named after Jacob Bernoulli who, in the late 1600s, studied them extensively). A binomial experiment takes place when the number of successes is counted in one or more Bernoulli Trials.

At ABC College, the withdrawal rate from an elementary physics course is 30% for any given term. This implies that, for any given term, 70% of the students stay in the class for the entire term. A “success” could be defined as an individual who withdrew. The random variable [latex]X=[/latex] the number of students who withdraw from the randomly selected elementary physics class.

A trainer is teaching a dolphin to do tricks. The probability that the dolphin successfully performs the trick is 35%, and the probability that the dolphin does not successfully perform the trick is 65%. Out of 20 attempts, you want to find the probability that the dolphin succeeds 12 times. State the probability question mathematically.

A fair coin is flipped 15 times. Each flip is independent. What is the probability of getting more than ten heads? Let [latex]X=[/latex] the number of heads in 15 flips of the fair coin. [latex]X[/latex] takes on the values 0, 1, 2, 3, …, 15. Since the coin is fair, [latex]p=0.5[/latex] and [latex]q=0.5[/latex]. The number of trials is [latex]n=15[/latex]. State the probability question mathematically.

A fair, six-sided die is rolled ten times. Each roll is independent. You want to find the probability of rolling a one more than three times. State the probability question mathematically.

This is a binomial problem because there is only a success or a __________, there are a fixed number of trials, and the probability of a success is 0.70 for each trial.
If we are interested in the number of students who do their homework on time, then how do we define [latex]X[/latex]?
What values does [latex]x[/latex] take on?
What is a “failure,” in words?
If [latex]p+q=1[/latex], then what is [latex]q[/latex]?
The words “at least” translate as what kind of inequality for the probability question [latex]P(x\geq40)[/latex].
[latex]X=[/latex] the number of statistics students who do their homework on time
0, 1, 2, …, 50
Failure is defined as a student who does not complete his or her homework on time. The probability of a success is [latex]p=0.70[/latex]. The number of trials is [latex]n=50[/latex].
[latex]q=0.30[/latex]
greater than or equal to (≥)The probability question is [latex]P(x\geq40)[/latex].

Binomial Distribution. Provided by : OpenStax. Located at : https://openstax.org/books/introductory-statistics/pages/4-3-binomial-distribution . License : CC BY: Attribution . License Terms : Access for free at https://openstax.org/books/introductory-statistics/pages/1-introduction
Introductory Statistics. Authored by : Barbara Illowsky, Susan Dean. Provided by : Open Stax. Located at : https://openstax.org/books/introductory-statistics/pages/1-introduction . License : CC BY: Attribution . License Terms : Access for free at https://openstax.org/books/introductory-statistics/pages/1-introduction

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That actually explain what's on your next test, binomial experiment, from class:, honors statistics.

A binomial experiment is a statistical experiment that meets the following criteria: 1) the experiment consists of a fixed number of trials, 2) each trial has only two possible outcomes, often referred to as 'success' and 'failure', 3) the probability of success is the same for each trial, and 4) the trials are independent of one another.

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5 Must Know Facts For Your Next Test

The number of trials in a binomial experiment is denoted by the variable $n$, and the probability of success in each trial is denoted by the variable $p$.
The binomial experiment assumes that the trials are independent, meaning the outcome of one trial does not affect the outcome of any other trial.
The random variable $X$ in a binomial experiment represents the number of successes in the $n$ trials, and $X$ follows a binomial distribution with parameters $n$ and $p$.
Binomial experiments are commonly used in fields such as quality control, market research, and clinical trials to study the proportion of successes in a fixed number of trials.
The binomial distribution can be used to calculate the probability of obtaining a certain number of successes in a binomial experiment.

Review Questions

A binomial experiment consists of a fixed number of independent trials, each with only two possible outcomes (usually referred to as 'success' and 'failure'), where the probability of success remains constant across all trials. This is in contrast to a Bernoulli trial, which is a single trial of a binomial experiment. In a binomial experiment, the random variable $X$ represents the number of successes in the $n$ trials, whereas in a Bernoulli trial, the random variable is a binary outcome (success or failure) for a single trial.
The binomial distribution is used to model the number of successes in a binomial experiment. The random variable $X$ in a binomial experiment follows a binomial distribution with parameters $n$ (the number of trials) and $p$ (the probability of success in each trial). The binomial distribution gives the probability of obtaining $x$ successes out of $n$ trials, where $x$ can take on values from 0 to $n$. The parameters $n$ and $p$ determine the shape and characteristics of the binomial distribution, with $n$ representing the number of trials and $p$ representing the probability of success in each trial.
The assumptions of independence and constant probability of success in a binomial experiment are crucial for the binomial distribution to accurately model the outcomes. The independence assumption ensures that the outcome of one trial does not affect the outcome of any other trial, allowing the probabilities to be multiplied together. The constant probability of success assumption ensures that the probability of success remains the same across all trials, which is necessary for the binomial distribution to accurately represent the possible outcomes and their corresponding probabilities. If these assumptions are violated, the binomial distribution may no longer be an appropriate model, and alternative statistical techniques may be required to analyze the experiment's results.

Related terms

Bernoulli Trial : A Bernoulli trial is a single trial of a binomial experiment with two possible outcomes, usually denoted as 'success' and 'failure', where the probability of success remains constant across all trials.

Binomial Distribution : The binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, where each trial has only two possible outcomes.

Random Variable : A random variable is a variable that can take on different values, each with a certain probability, in the context of a statistical experiment.

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Binomial Probability Distribution

To understand binomial distributions and binomial probability, it helps to understand binomial experiments and some associated notation; so we cover those topics first.

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Binomial Experiment

A binomial experiment is a statistical experiment that has the following properties:

The experiment consists of n repeated trials.
Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.
The probability of success, denoted by P , is the same on every trial.
The trials are independent ; that is, the outcome on one trial does not affect the outcome on other trials.

Consider the following statistical experiment. You flip a coin 2 times and count the number of times the coin lands on heads. This is a binomial experiment because:

The experiment consists of repeated trials. We flip a coin 2 times.
Each trial can result in just two possible outcomes - heads or tails.
The probability of success is constant - 0.5 on every trial.
The trials are independent; that is, getting heads on one trial does not affect whether we get heads on other trials.

The following notation is helpful, when we talk about binomial probability.

x : The number of successes that result from the binomial experiment.
n : The number of trials in the binomial experiment.
P : The probability of success on an individual trial.
Q : The probability of failure on an individual trial. (This is equal to 1 - P .)
n! : The factorial of n (also known as n factorial).
b( x ; n, P ): Binomial probability - the probability that an n -trial binomial experiment results in exactly x successes, when the probability of success on an individual trial is P .
n C r : The number of combinations of n things, taken r at a time.

Binomial Distribution

A binomial random variable is the number of successes x in n repeated trials of a binomial experiment. The probability distribution of a binomial random variable is called a binomial distribution .

Suppose we flip a coin two times and count the number of heads (successes). The binomial random variable is the number of heads, which can take on values of 0, 1, or 2. The binomial distribution is presented below.

Number of heads	Probability
0	0.25
1	0.50
2	0.25

The binomial distribution has the following properties:

The mean of the distribution (μ x ) is equal to n * P .
The variance (σ 2 x ) is n * P * ( 1 - P ).
The standard deviation (σ x ) is sqrt[ n * P * ( 1 - P ) ].

Binomial Formula and Binomial Probability

The binomial probability refers to the probability that a binomial experiment results in exactly x successes. For example, in the above table, we see that the binomial probability of getting exactly one head in two coin flips is 0.50.

Given x , n , and P , we can compute the binomial probability based on the binomial formula:

Binomial Formula. Suppose a binomial experiment consists of n trials and results in x successes. If the probability of success on an individual trial is P , then the binomial probability is:

b( x ; n, P ) = n C x * P x * (1 - P) n - x or b( x ; n, P ) = { n! / [ x! (n - x)! ] } * P x * (1 - P) n - x

Example 1 Suppose a die is tossed 5 times. What is the probability of getting exactly 2 fours?

Solution: This is a binomial experiment in which the number of trials is equal to 5, the number of successes is equal to 2, and the probability of success on a single trial is 1/6 or about 0.167. Therefore, the binomial probability is:

b(2; 5, 0.167) = 5 C 2 * (0.167) 2 * (0.833) 3 b(2; 5, 0.167) = 10 * (0.167) 2 * (0.833) 3 b(2; 5, 0.167) = 0.161

Example 2 What is the probability that the world series will last 4 games? 5 games? 6 games? 7 games? Assume that the teams are evenly matched.

Solution: The solution to this problem requires a creative application of the binomial formula. If you can follow the logic of this solution, you have a good understanding of the material covered in the tutorial, to this point.

In the world series, there are two baseball teams. The series ends when the winning team wins 4 games. Therefore, we define a success as a win by the team that ultimately becomes the world series champion.

For the purpose of this analysis, we assume that the teams are evenly matched. Therefore, the probability that a particular team wins a particular game is 0.5.

Let's look first at the simplest case. What is the probability that the series lasts only 4 games. This can occur if one team wins the first 4 games. The probability of the National League team winning 4 games in a row is:

b(4; 4, 0.5) = 4 C 4 * (0.5) 4 * (0.5) 0 = 0.0625

Similarly, when we compute the probability of the American League team winning 4 games in a row, we find that it is also 0.0625. Therefore, probability that the series ends in four games would be 0.0625 + 0.0625 = 0.125; since the series would end if either the American or National League team won 4 games in a row.

Now let's tackle the question of finding probability that the world series ends in 5 games. The trick in finding this solution is to recognize that the series can only end in 5 games, if one team has won 3 out of the first 4 games. So let's first find the probability that the American League team wins exactly 3 of the first 4 games.

b(3; 4, 0.5) = 4 C 3 * (0.5) 3 * (0.5) 1 = 0.25

Okay, here comes some more tricky stuff, so listen up. Given that the American League team has won 3 of the first 4 games, the American League team has a 50/50 chance of winning the fifth game to end the series. Therefore, the probability of the American League team winning the series in 5 games is 0.25 * 0.50 = 0.125. Since the National League team could also win the series in 5 games, the probability that the series ends in 5 games would be 0.125 + 0.125 = 0.25.

The rest of the problem would be solved in the same way. You should find that the probability of the series ending in 6 games is 0.3125; and the probability of the series ending in 7 games is also 0.3125.

Cumulative Binomial Probability

A cumulative binomial probability refers to the probability that the binomial random variable falls within a specified range (e.g., is greater than or equal to a stated lower limit and less than or equal to a stated upper limit).

To compute a cumulative binomial probability, we find the sum of relevant individual binomial probabilities, as illustrated in the examples below.

Example 3 The probability that a student is accepted to a prestigious college is 0.3. If 5 students from the same school apply, what is the probability that at most 2 are accepted?

Solution: To solve this problem, we compute 3 individual probabilities, using the binomial formula. The sum of all these probabilities is the answer we seek. Thus,

b(x < 2; 5, 0.3) = b(x = 0; 5, 0.3) + b(x = 1; 5, 0.3) + b(x = 2; 5, 0.3) b(x < 2; 5, 0.3) = 0.1681 + 0.3601 + 0.3087 b(x < 2; 5, 0.3) = 0.8369

Example 4 What is the probability of obtaining 45 or fewer heads in 100 tosses of a coin?

Solution: To solve this problem, we compute 46 individual binomial probabilities, using the binomial formula. The sum of all these binomial probabilities is the answer we seek. Thus,

b(x < 45; 100, 0.5) = b(x = 0; 100, 0.5) + b(x = 1; 100, 0.5) + . . . + b(x = 44; 100, 0.5) + b(x = 45; 100, 0.5) b(x < 45; 100, 0.5) = 0.184

Binomial Calculator

As you may have noticed, the binomial formula requires many time-consuming computations. The Binomial Calculator can do this work for you - quickly, easily, and error-free. Use the Binomial Calculator to compute binomial probabilities and cumulative binomial probabilities. The calculator is free. It can found in the Stat Trek main menu under the Stat Tools tab. Or you can tap the button below.

Here is the solution to this problem, using the Binomial Calculator :

Example 5 Suppose it were possible to take a simple random sample of 120 newborns. Find the probability that no more than 40% will be boys. Assume equal probabilities for the births of boys and girls.

Solution: We know that 40% of 120 is 48. Therefore, we want to know the probability that a random sample of 120 newborns will include no more than 48 boys. The solution to this problem requires that we compute the following cumulative binomial probability.

b(x < 48; 120, 0.5) = b(x = 0; 120, 0.5) + b(x = 1; 120, 0.5) + ... + b(x = 48; 120, 0.5) b(x < 48; 120, 0.5) = 0.0 + 0.0 + ... + 0.00662 b(x < 48; 120, 0.5) = 0.01766

Note: Finding this cumulative binomial probability requires requires computing 49 individual binomial probabilities. It can be done by hand, but it is much easier to use the Binomial Calculator , as shown below:

Math Article

Binomial Distribution

In probability theory and statistics, the binomial distribution is the discrete probability distribution that gives only two possible results in an experiment, either Success or Failure . For example, if we toss a coin, there could be only two possible outcomes: heads or tails, and if any test is taken, then there could be only two results: pass or fail. This distribution is also called a binomial probability distribution.

There are two parameters n and p used here in a binomial distribution. The variable ‘n’ states the number of times the experiment runs and the variable ‘p’ tells the probability of any one outcome. Suppose a die is thrown randomly 10 times, then the probability of getting 2 for anyone throw is ⅙. When you throw the dice 10 times, you have a binomial distribution of n = 10 and p = ⅙. Learn the formula to calculate the two outcome distribution among multiple experiments along with solved examples here in this article.

Table of Contents:

Negative Binomial Distribution

Mean and Variance

Binomial Distribution Vs Normal Distribution

Solved Problems

Practice Problems

Binomial probability distribution.

In binomial probability distribution, the number of ‘Success’ in a sequence of n experiments, where each time a question is asked for yes-no, then the boolean-valued outcome is represented either with success/yes/true/one (probability p) or failure/no/false/zero (probability q = 1 − p). A single success/failure test is also called a Bernoulli trial or Bernoulli experiment, and a series of outcomes is called a Bernoulli process . For n = 1, i.e. a single experiment, the binomial distribution is a Bernoulli distribution . The binomial distribution is the base for the famous binomial test of statistical importance.

In probability theory and statistics, the number of successes in a series of independent and identically distributed Bernoulli trials before a particularised number of failures happens. It is termed as the negative binomial distribution. Here the number of failures is denoted by ‘r’. For instance, if we throw a dice and determine the occurrence of 1 as a failure and all non-1’s as successes. Now, if we throw a dice frequently until 1 appears the third time, i.e., r = three failures, then the probability distribution of the number of non-1s that arrived would be the negative binomial distribution.

Binomial Distribution Examples

As we already know, binomial distribution gives the possibility of a different set of outcomes. In real life, the concept is used for:

Finding the quantity of raw and used materials while making a product.
Taking a survey of positive and negative reviews from the public for any specific product or place.
By using the YES/ NO survey, we can check whether the number of persons views the particular channel.
To find the number of male and female employees in an organisation.
The number of votes collected by a candidate in an election is counted based on 0 or 1 probability.

Also, read:

Binomial Distribution Formula

The binomial distribution formula is for any random variable X, given by;

P(x:n,p) = C p (1-p)

P(x:n,p) = C p (q)

n = the number of experiments

x = 0, 1, 2, 3, 4, …

p = Probability of Success in a single experiment

q = Probability of Failure in a single experiment = 1 – p

The binomial distribution formula can also be written in the form of n-Bernoulli trials, where n C x = n!/x!(n-x)!. Hence,

P(x:n,p) = n!/[x!(n-x)!].p x .(q) n-x

Binomial Distribution Mean and Variance

For a binomial distribution, the mean, variance and standard deviation for the given number of success are represented using the formulas

Mean, μ = np

Variance, σ 2 = npq

Standard Deviation σ= √(npq)

Where p is the probability of success

q is the probability of failure, where q = 1-p

The main difference between the binomial distribution and the normal distribution is that binomial distribution is discrete, whereas the normal distribution is continuous. It means that the binomial distribution has a finite amount of events, whereas the normal distribution has an infinite number of events. In case, if the sample size for the binomial distribution is very large, then the distribution curve for the binomial distribution is similar to the normal distribution curve.

Properties of Binomial Distribution

The properties of the binomial distribution are:

There are two possible outcomes: true or false, success or failure, yes or no.
There is ‘n’ number of independent trials or a fixed number of n times repeated trials.
The probability of success or failure remains the same for each trial.
Only the number of success is calculated out of n independent trials.
Every trial is an independent trial, which means the outcome of one trial does not affect the outcome of another trial.

Binomial Distribution Examples And Solutions

Example 1: If a coin is tossed 5 times, find the probability of:

(a) Exactly 2 heads

(b) At least 4 heads.

(a) The repeated tossing of the coin is an example of a Bernoulli trial. According to the problem:

Number of trials: n=5

Probability of head: p= 1/2 and hence the probability of tail, q =1/2

For exactly two heads:

P(x=2) = 5 C2 p 2 q 5-2 = 5! / 2! 3! × (½) 2 × (½) 3

P(x=2) = 5/16

(b) For at least four heads,

x ≥ 4, P(x ≥ 4) = P(x = 4) + P(x=5)

P(x = 4) = 5 C4 p 4 q 5-4 = 5!/4! 1! × (½) 4 × (½) 1 = 5/32

P(x = 5) = 5 C5 p 5 q 5-5 = (½) 5 = 1/32

P(x ≥ 4) = 5/32 + 1/32 = 6/32 = 3/16

Example 2: For the same question given above, find the probability of:

a) Getting at most 2 heads

Solution: P (at most 2 heads) = P(X ≤ 2) = P (X = 0) + P (X = 1) + + P (X = 2)

P(X = 0) = (½) 5 = 1/32

P(X=1) = 5 C 1 (½) 5. = 5/32

P(x=2) = 5 C2 p 2 q 5-2 = 5! / 2! 3! × (½) 2 × (½) 3 = 5/16

P(X ≤ 2) = 1/32 + 5/32 + 5/16 = 1/2

A fair coin is tossed 10 times, what are the probability of getting exactly 6 heads and at least six heads.

Let x denote the number of heads in an experiment.

Here, the number of times the coin tossed is 10. Hence, n=10.

The probability of getting head, p ½

The probability of getting a tail, q = 1-p = 1-(½) = ½.

The binomial distribution is given by the formula:

P(X= x) = n C x p x q n-x , where = 0, 1, 2, 3, …

Therefore, P(X = x) = 10 C x (½) x (½) 10-x

(i) The probability of getting exactly 6 heads is:

P(X=6) = 10 C 6 (½) 6 (½) 10-6

P(X= 6) = 10 C 6 (½) 10

P(X = 6) = 105/512.

Hence, the probability of getting exactly 6 heads is 105/512.

(ii) The probability of getting at least 6 heads is P(X ≥ 6)

P(X ≥ 6) = P(X=6) + P(X=7) + P(X= 8) + P(X = 9) + P(X=10)

P(X ≥ 6) = 10 C 6 (½) 10 + 10 C 7 (½) 10 + 10 C 8 (½) 10 + 10 C 9 (½) 10 + 10 C 10 (½) 10

P(X ≥ 6) = 193/512.

Solve the following problems based on binomial distribution:

The mean and variance of the binomial variate X are 8 and 4 respectively. Find P(X<3).
The binomial variate X lies within the range {0, 1, 2, 3, 4, 5, 6}, provided that P(X=2) = 4P(x=4). Find the parameter “p” of the binomial variate X.
In binomial distribution, X is a binomial variate with n= 100, p= ⅓, and P(x=r) is maximum. Find the value of r.

Frequently Asked Questions on Binomial Distribution

What is meant by binomial distribution.

The binomial distribution is the discrete probability distribution that gives only two possible results in an experiment, either success or failure.

Mention the formula for the binomial distribution.

The formula for binomial distribution is: P(x: n,p) = n C x p x (q) n-x Where p is the probability of success, q is the probability of failure, n= number of trials

What is the formula for the mean and variance of the binomial distribution?

The mean and variance of the binomial distribution are: Mean = np Variance = npq

What are the criteria for the binomial distribution?

The number of trials should be fixed. Each trial should be independent. The probability of success is exactly the same from one trial to the other trial.

What is the difference between a binomial distribution and normal distribution?

The binomial distribution is discrete, whereas the normal distribution is continuous.

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Binomial Theorem: Simple Definition, Formula, Step by Step Videos

What is the binomial theorem.

Basic Information on the Binomial Theorem

What is a Bernoulli Trial?
Binomial Distribution
Variance of Binomial Distribution.
Standard deviation for a binomial distribution.
Mean of Binomial Distribution.
Negative Binomial Experiment .
Binomial Experiment: Is it one or not?
Binomial Formula .
What is Bernoulli Sampling?

Binomial Theorem: More Advanced Topics

Binomial Confidence Intervals.
Using the Normal Approximation to solve a Binomial Problem.
The Binomial Hypothesis Test .
The Success/Failure Condition .
Binomial Distribution Calculator
Binomial Probability in Minitab .
Binomial Distribution Table.
How to Read a Binomial Distribution Table.

Binomials in Real Life

You find a parking space, or you don’t.
You win Powerball, or you don’t.
You get a birthday gift from your partner, or you don’t.
A drug cures, or it doesn’t.
A computer works, or it doesn’t.
You get a paycheck, or you don’t.

You’d want to know if a computer is going to work before you buy it. Which means that you’ve probably used binomials without realizing it. Before you buy, you might have checked reviews. While a lot of review sites have ratings, they’re mostly in two categories: it’s great, or it sucks. Whether something works or not becomes more important in life or death situations; For example, taking a drug to see if it cures terminal disease. In fact, “life” and “death” are two outcomes, which makes anything that could result in death a binomial.

Of course, you need a little math to describe it. A binomial distribution is the probability of something happening in an experiment. There are two variables, n and p.

n: The number of times the experiment happens.
p: The probability of one specific outcome.

For example, if you roll a die 20 times, then “n” is 20. If you buy ten lottery tickets, then n is 10. The odds of rolling a number (like 4) are 1 out of 6. So p=(1/6). And the odds of you winning a lottery are usually high, maybe one out of a million. In that case, p would equal (1/1 million).

There are hundreds of ways you could measure success, but this is one of the simplest. Something works, or it doesn’t. It isn’t enough to say “maybe it will work”, especially if there’s money involved. For example, drugs cost billions to develop. Some medication cost patients over $50,000 a year. Before the drug is made, the manufacturer wants to know if it will work. Before you a buy a drug, you’ll want to know if it’s going to work. They can all be measured with binomials.

Imagine a life without binomials. You wouldn’t know what the chances are of a drug curing you. You wouldn’t know what the odds were that you would get a side effect. You wouldn’t be able to compare different drugs. And you probably wouldn’t even be able to buy it in the first place. Why? Because buying something is a binomial experiment too. Can I afford it or can’t it? Do I have money in my wallet or don’t I? Do I have enough gas in my car? And so on…!

Binomial Expansions and Series

The binomial formula in statistics is mostly used for counting and for calculating probabilities in experiments. A very similar technique, called binomial series expansion , is used in calculus for rewriting complicated functions into a simpler (binomial) form.

Check out our YouTube channel ! We’ve got hundreds of videos up on all kinds of stats topics. If you’re taking a class, subscribe. Comments and suggestions are always welcome.

Have a question not covered here? Use the Site Search at the top of the page to find what you’re looking for.

Agresti A. (1990) Categorical Data Analysis. John Wiley and Sons, New York. Gonick, L. (1993). The Cartoon Guide to Statistics . HarperPerennial. Kotz, S.; et al., eds. (2006), Encyclopedia of Statistical Sciences , Wiley. Everitt, B. S.; Skrondal, A. (2010), The Cambridge Dictionary of Statistics , Cambridge University Press. Vogt, W.P. (2005). Dictionary of Statistics & Methodology: A Nontechnical Guide for the Social Sciences . SAGE. Lindstrom, D. (2010). Schaum’s Easy Outline of Statistics , Second Edition (Schaum’s Easy Outlines) 2nd Edition. McGraw-Hill Education

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Statistics By Jim

Making statistics intuitive

Binomial Distribution Formula: Probability, Standard Deviation & Mean

By Jim Frost 2 Comments

Binomial Distribution Formula

Use the binomial distribution formula to calculate the likelihood an event will occur a specific number of times in a set number of opportunities. I’ll show you the binomial distribution formula to calculate these probabilities manually.

In this post, I’ll walk you through the formulas for how to find the probability, mean, and standard deviation of the binomial distribution and provide worked examples.

Note that this post focuses on the binomial distribution formulas and calculations. For more information about the distribution itself and how to use and graph it, please read Binomial Distribution: Uses & Calculator .

The binomial distribution formula for probabilities is the following:

Binomial distribution formula that applies to a binomial random variable.

n is the number of trials.
x is the number of successes
p is the probability of a success.
(1–p) is the chance of failure.

Use this formula to calculate the binomial probability for X successes occurring in n trials.

nCx is the number of ways to obtain samples with the specified number of successes occurring within the set number of trials where the order of outcomes does not matter. Specifically, it’s the number of combinations without repetition. For more information, read my post about Finding Combinations .

The binomial distribution formula takes the number of combinations, multiplies that by the probability of success raised by the number of successes, and multiplies that by the probability of failures raised by the number of failures.

Let’s work through an example calculation to bring the formula to life!

Worked Example of Finding a Binomial Probability

We’ll use the binomial distribution formula to calculate the chances of rolling exactly three sixes in ten die rolls for this example. Here are the values to enter into the formula:

(1–p) = 0.8333

For the number of combinations, we have:

Example calculations for the number of combinations.

Now, let’s enter our values into the binomial distribution formula.

Worked example of using the binomial distribution formula to calculate probabilities for a random binomial variable.

This calculation finds that the binomial probability of rolling three 6s in 10 rolls is 0.1540.

If you need to calculate a cumulative probability for a binomial random variable, calculate the likelihood for each individual outcome and then sum them for all outcomes of interest.

For example, if you want to calculate the probability of ≥ 3 sixes in 10 rolls, calculate the likelihoods for three sixes, four sixes, etc., on up to ten sixes. Then sum that set of binomial probabilities.

Read on to learn about the formulas to calculate the mean and standard deviation of the binomial distribution!

Expected Value of Binomial Distribution

The expected value of the binomial distribution is its mean.

The binomial distribution formula for the expected value is the following:

Multiply the number of trials (n) by the success probability (p). This value represents the average or expected number of successes.

For example, we roll the die ten times, and the probability of rolling a six is 0.1667. Let’s enter these values into the formula

10 * 0.1667

The mean for this binomial distribution is 1.667. On average, we’d expect to roll that many sixes in ten rolls. Of course, the actual counts of successes will always be either zero or a positive integer.

This mean is the expected value for a binomial distribution. Learn more about Expected Values: Definition, Formula & Finding .

Standard Deviation of the Binomial Distribution

The binomial distribution formula for the standard deviation is the following:

As before, n and p are the number of trials and success probability, respectively. (1 – p) is the likelihood of failure.

Notice that the standard deviation of the binomial distribution is at its maximum when the probabilities for success and failure are both 0.5. As those probabilities move away from 0.5 in opposite directions, it decreases. Additionally, it increases as the number of trials (n) increase.

For our die example we have n = 10 rolls, a success probability of p = 0.1667, and a failure probability of (1 – p) = 0.833. Let’s enter these values into the formula.

10 * 0.1667 * 0.8333 = 1.3891. That’s the variance , which uses squared units.

To find the standard deviation of the binomial distribution, we need to take the square root of the variance.

The standard deviation represents the variability of the probabilities around the mean of the binomial distribution. Learn more about the Standard Deviation .

By using the formula for the binomial distribution, it is easy to calculate its probabilities, means, and standard deviations.

Reader Interactions

June 13, 2024 at 2:31 am

I would like to have a question regarding the distribution of outcomes of multiple binomial distributions.

I have a sample of 50 subjects, where every subject completes a task with two possible outcomes (left or right hand use, with 50% probability) 30 times. On an individual level, this leads to a binomial distribution of outcomes for each subject, where a z-score can be calculated for each subject to see how far the hand uses are from the expected 50% (15 left and right hand use) level. Based on the z-scores, we deem those with larger values than 1.96 (›1.96) as right-handed, those with lower values than -1.96 (‹-1.96) as left-handed and those with values in between (-1.96‹‹1.96) as ambilateral.

Question: what is the expected ratio of z-scores (or distribution of z-scores) in my sample of 50 subjects? Is it the same as at the individual level with 95% chance of being ambilateral and 2.5-2.5% as being left or right-handed?

December 14, 2023 at 5:30 pm

I have a probability question I have been trying to figure out. Here is the scenario: A group (12) of golfers gather each Saturday to play in a match. Each player is trying to gain points based on their individual handicap. For example, my handicap is 34 and if I get 34 pts I finish Even for the day. If I get 32 pts, I finish -2; 36 pts, +2 and so on.

At the end of the match players are randomly drawn as teammates from a hat. The team with the highest net score wins the match. Let’s assume for this match, teams will be drawn out in 2s. Let’s also assume that the maximum + or – for each player is 5. For instance, I may be drawn with player #7 who is +4 for the day and I am -1 for the day. Our team score is +3.

What is the probability that I (my team) will win? Do my odds change based on my score?

What if there are 20 players? Or, if teams are divided into 4s?

Thanks for providing some help and insight!

Comments and Questions Cancel reply

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What Is Binomial Distribution?

How It Works

The Bottom Line

Corporate Finance
Financial Analysis

Binomial Distribution: Definition, Formula, Analysis, and Example

Investopedia contributors come from a range of backgrounds, and over 25 years there have been thousands of expert writers and editors who have contributed.

probability and statistics binomial experiment definition

Investopedia / Eliana Rodgers

Binomial distribution is a statistical distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters or assumptions.

The underlying assumptions of binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive or independent of one another.

Key Takeaways

Binomial distribution is a statistical probability distribution that summarizes the likelihood that a value will take one of two independent values under a given set of parameters or assumptions.
The underlying assumptions of binomial distribution are that there is only one outcome for each trial, that each trial has the same probability of success, and that each trial is mutually exclusive or independent of one another.
Binomial distribution is a common discrete distribution used in statistics, as opposed to a continuous distribution, such as normal distribution.

Understanding Binomial Distribution

To start, the “binomial” in binomial distribution means two terms—the number of successes and the number of attempts. Each is useless without the other.

Binomial distribution is a common discrete distribution used in statistics, as opposed to a continuous distribution, such as normal distribution . This is because binomial distribution only counts two states, typically represented as 1 (for a success) or 0 (for a failure), given a number of trials in the data. Binomial distribution thus represents the probability for x successes in n trials, given a success probability p for each trial.

Binomial distribution summarizes the number of trials, or observations, when each trial has the same probability of attaining one particular value. Binomial distribution determines the probability of observing a specific number of successful outcomes in a specified number of trials.

Binomial distribution is often used in social science statistics as a building block for models for dichotomous outcome variables, such as whether a Republican or Democrat will win an upcoming election, whether an individual will die within a specified period of time, etc. It also has applications in finance, banking, and insurance, among other industries.

Analyzing Binomial Distribution

A binomial distribution's expected value, or mean, is calculated by multiplying the number of trials (n) by the probability of successes (p), or n × p.

For example, the expected value of the number of heads in 100 trials of heads or tails is 50, or (100 × 0.5). Another common example of binomial distribution is estimating the chances of success for a free-throw shooter in basketball, where 1 = a basket made and 0 = a miss.

The binomial distribution function is calculated as:

P ( x : n , p ) = n C x p x ( 1 - p ) n - x

n is the number of trials (occurrences)
x is the number of successful trials
p is the probability of success in a single trial
n C x is the combination of n and x. A combination is the number of ways to choose a sample of x elements from a set of n distinct objects where order does not matter, and replacements are not allowed. Note that n C x = n! / r! ( n − r ) ! ), where ! is factorial (so, 4! = 4 × 3 × 2 × 1).

The mean of the binomial distribution is np, and the variance of the binomial distribution is np (1 − p). When p = 0.5, the distribution is symmetric around the mean—such as when flipping a coin because the chances of getting heads or tails is 50%, or 0.5. When p > 0.5, the distribution curve is skewed to the left. When p < 0.5, the distribution curve is skewed to the right.

The binomial distribution is the sum of a series of multiple independent and identically distributed Bernoulli trials. In a Bernoulli trial, the experiment is said to be random and can only have two possible outcomes: success or failure.

For instance, flipping a coin is considered to be a Bernoulli trial ; each trial can only take one of two values (heads or tails), each success has the same probability, and the results of one trial do not influence the results of another. Bernoulli distribution is a special case of binomial distribution where the number of trials n = 1.

Example of Binomial Distribution

Binomial distribution is calculated by multiplying the probability of success raised to the power of the number of successes and the probability of failure raised to the power of the difference between the number of successes and the number of trials. Then, multiply the product by the combination of the number of trials and successes.

For example, assume that a casino created a new game in which participants can place bets on the number of heads or tails in a specified number of coin flips. Assume a participant wants to place a $10 bet that there will be exactly six heads in 20 coin flips. The participant wants to calculate the probability of this occurring, and therefore, they use the calculation for binomial distribution.

The probability was calculated as (20! / (6! × (20 - 6)!)) × (0.50) (6) × (1 - 0.50) (20 - 6) . Consequently, the probability of exactly six heads occurring in 20 coin flips is 0.0369, or 3.7%. The expected value was 10 heads in this case, so the participant made a poor bet. The graph below shows that the mean is 10 (the expected value), and the chances of getting six heads is on the left tail in red. You can see that there is less of a chance of six heads occurring than seven, eight, nine, 10, 11, 12, or 13 heads.

StatCrunch Binomial Calculator

So how can this be used in finance? One example: Let’s say you’re a bank, a lender , that wants to know within three decimal places the likelihood of a particular borrower defaulting. What are the chances of so many borrowers defaulting that they would render the bank insolvent? Once you use the binomial distribution function to calculate that number, you have a better idea of how to price insurance and, ultimately, how much money to lend out and keep in reserve.

Binomial distribution is a statistical probability distribution that states the likelihood that a value will take one of two independent values under a given set of parameters or assumptions.

How Is Binomial Distribution Used?

This distribution pattern is used in statistics but has implications in finance and other fields. Banks may use it to estimate the likelihood of a particular borrower defaulting, how much money to lend, and the amount to keep in reserve. It’s also used in the insurance industry to determine policy pricing and assess risk .

Why Is Binomial Distribution Important?

Binomial distribution is used to figure the probability of a pass or fail outcome in a survey, or experiment replicated numerous times. There are only two potential outcomes for this type of distribution. More broadly, distribution is an important part of analyzing data sets to estimate all the potential outcomes of the data and how frequently they occur. Forecasting and understanding the success or failure of outcomes is essential to business development.

The binomial distribution is an important statistical distribution that describes binary outcomes (such as the flip of a coin, a yes/no answer, or an on/off condition). Understanding its characteristics and functions is important for data analysis in various contexts that involve an outcome taking one of two independent values.

It has applications in social science, finance, banking, insurance, and other areas. For instance, it can be used to estimate whether a borrower will default on a loan, whether an options contract will finish in-the-money or out-of-the-money, or whether a company will miss or beat earnings estimates.

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Binomial Distribution

In statistics and probability theory, the binomial distribution is the probability distribution that is discrete and applicable to events having only two possible results in an experiment, either success or failure. (the prefix “bi” means two, or twice). A few circumstances where we have binomial experiments are tossing a coin: head or tail, the result of a test: pass or fail, selected in an interview: yes/ no, or nature of the product: defective/non-defective. Such a distribution of a binomial random variable is called a binomial probability distribution.

Binomial Distribution is a commonly used discrete distribution in statistics. The normal distribution as opposed to a binomial distribution is a continuous distribution. Let us learn the formula to calculate the Binomial distribution considering many experiments and a few solved examples for a better understanding.

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What Is Binomial Distribution?

The binomial distribution is the probability distribution of a binomial random variable. A random variable is a real-valued function whose domain is the sample space of a random experiment. Let us consider an example to understand this better.

Toss a fair coin twice. This is a binomial experiment. There are 4 possible outcomes of this experiment. {HH, HT, TH, TT}. Consider getting one head as the success. Count the number of successes in each possible outcome. Here n(getting heads) is the success in n repeated trials of a binomial experiment.. n(X) = 0, 1, or 2 is the binomial random variable . The distribution of probability is of a binomial random variable, and this is known as a binomial distribution.

No. of heads(n(X))	Probability of getting a head(P(X))
0	P(x = 0) = 1/4 = 0.25
1	P(x = 1) = P(HT) = 1/4 + 1/4 = 0.50
2	P(x = 2) = P(HH) = 1/4 = 0.25

This table shows that getting one head in a single flip is 0.50. Now if a coin is flipped 3 times, consider we are intended to find the binomial distribution of getting two heads. Tossing 3 coins result in 8 outcomes. {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. The probability of getting two heads [P(HH)] is 3/8. Similarly, we can calculate the probability of getting one head, 2 heads, and 3 heads and 0 heads. The binomial probability distribution is given in terms of a random variable as:

P(X = 0) = 1/8

P(X = 1) = 3/8

P(X = 2) = 3/8

P(X = 3)= 1/8

Binomial Distribution in Statistics

The binomial distribution forms the base for the famous binomial test of statistical importance. The binomial distribution represents the probability for 'x' successes of an experiment in 'n' trials, given a success probability 'p' for each trial at the experiment. Two parameters n and p are used here in the binomial distribution. The variable ‘n’ represents the number of trials and the variable ‘p’ states the probability of any one(each) outcome. A test that has a single outcome such as success/failure is also called a Bernoulli trial or Bernoulli experiment, and a series of outcomes is called a Bernoulli process.

Consider an experiment where each time a question is asked for a yes/no with a series of n experiments. Then in the binomial probability distribution, the boolean-valued outcome the success/yes/true/one is represented with probability p and the failure/no/false/zero with probability q (q = 1 − p). In a single experiment when n = 1, the binomial distribution is called a Bernoulli distribution .

If a die is thrown randomly 10 times, then the probability of getting a 3 for any throw is 1/6. Similarly, if we throw the dice 10 times, we have n = 10 and p = 1/6, q = 5/6

Negative Binomial Distribution

Let's understand with an example when can a binomial distribution be negative. Suppose we throw a die and determine that the occurrence of 2 will be a failure and all non-2’s will be successes. Let the failures be denoted by ‘r’. Now, if the die is thrown frequently until 2 appears the third time, i.e., r = three failures, then the binomial distribution of the number of non-2's that arrived would be the negative binomial distribution .

Binomial Distribution Formula

The binomial distribution formula is for any random variable X, given by; P(x:n,p) = n C x p x (1-p) n-x Or P(x:n,p) = n C x p x (q) n-x

Where p is the probability of success, q is the probability of failure, and n = number of trials. The binomial distribution formula is also written in the form of n-Bernoulli trials.

where n C x = n!/x!(n-x)!. Hence, P(x:n,p) = n!/[x!(n-x)!].p x .(q) n-x

Binomial Distribution Calculation

The image given below shows the formula used for binomial distribution calculation:

Application of Binomial Distribution

We now already know that binomial distribution gives the probability of a different set of outcomes. In real life, the concept of the binomial distribution is used for:

Finding the quantity of raw and used materials while making a product.
Taking a survey of positive and negative reviews from the public for any specific product or place.
By using the YES/ NO survey
To find the number of male and female students in a university.
The number of votes collected by a candidate in an election is counted based on 0 or 1 probability.

Consider a card selected at a random and replaced. If this experiment is repeated 5 times, let us find the probability of selecting exactly 3 hearts. Let us determine the number of trials, success, and the failure. The trial is the drawing a card 5 times. Thus n = 5.

success: card drawn is a heart = p = 1/4 = 0.25

failure: card drawn is not a heart = q = 1-0.25 = 0.75

Using the binomial distribution formula, we get 5 C $_3$ (0,25) 3 (0.75) 2 = 0.088

Binomial Distribution Mean and Variance

For a binomial distribution, the mean , variance and standard deviation for the given number of success are represented using the formulas

Mean, μ = np
Variance, σ 2 = npq
Standard Deviation σ= √(npq)

Where p is the probability of success q is the probability of failure, where q = 1-p

Binomial Distribution Vs Normal Distribution

The main difference between the binomial distribution and the normal distribution is that the binomial distribution is discrete, whereas the normal distribution is continuous. It means that the binomial distribution has a finite amount of events, whereas the normal distribution has an infinite number of events. In case, if the sample size for the binomial distribution is very large, then the distribution curve for the binomial distribution is similar to the normal distribution curve.

Properties of Binomial Distribution

The properties of the binomial distribution are:

There are only two distinct possible outcomes: true/false, success/failure, yes/no.
There is a fixed number of 'n' times repeated trials in a given experiment.
The probability of success or failure remains constant for each attempt/trial.
Only the successful attempts are calculated out of 'n' independent trials.
Every trial is an independent trial on its own, this means that the outcome of one trial has no effect on the outcome of another trial.

Important Notes on Binomial Distribution

For using the binomial distribution, the number of observations or trials in an experiment is fixed or finite.
Each observation/attempt/trial is independent on its own. This means none of the trials have an effect on the probability of the next trial.
Each trial has an equal probability of occurrence. The probability of success is exactly the same from one trial to another.

☛ Related Articles:

Normal Distribution Formula
Cumulative Frequency
Frequency Distribution

Binomial Distribution Examples

Example 1: If a coin is tossed 5 times, using binomial distribution find the probability of:

(a)Exactly 2 heads

(b) At least 4 heads.

(a) The repeated tossing of the coin is an example of a Bernoulli trial. According to the problem:

Number of trials: n=5

Probability of head: p= 1/2 and hence the probability of tail, q =1/2

For exactly two heads:

Using binomial distribution formula,

P(x=2) = 5 C 2 p 2 q 5-2 = 5! / 2! 3! × (½) 2 × (½) 3

P(x=2) = 5/16

(b) For at least four heads,

x ≥ 4, P(x ≥ 4) = P(x = 4) + P(x=5)

Hence, using binomial distribution formula,

P(x = 4) = 5 C 4 p 4 q 5-4 = 5!/4! 1! × (½) 4 × (½) 1 = 5/32

P(x = 5) = 5 C 5 p 5 q 5-5 = (½) 5 = 1/32

Answer: Therefore, P(x ≥ 4) = 5/32 + 1/32 = 6/32 = 3/16

Example 2: For the same question given above, using the binomial distribution find the probability of getting at most 2 heads.

Solution: P(at most 2 heads) = P(X ≤ 2) = P (X = 0) + P (X = 1)

P(X = 0) = (½)5 = 1/32

Using binomial distribution formula, we get

P(X=1) = 5 C 1 (½) 5 = 5/32

Answer: Therefore, P(X ≤ 2) = 1/32 + 5/32 = 3/16

Example 3: A random variable X has the following binomial distribution. Determine P(X>6) and P(0<X<3)

X	0	1	2	3	4	5	6	7
P(X)	0	k	2k	2k	3k	k	2k	7k + k

This is a binomial distribution.

To find k. The sum of all the probabilities = 1

0 + k + 2k +2k + 3k + k 2 + 2k 2 + 7k 2 + k = 1

10k 2 + 8 k = 1

Solving for k , we get k = 0.1 and -1, We consider k = 0.1 as k = -1 makes the probability negative which is not possible.

i) P(X>6)= 7k 2 + k

7(0.1) 2 + 0.1

ii) P(0<X<3)

= k + 2k = 3k

Answer: P(X>6)= 0.17 and P(0<X<3) = 0.3

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Binomial Distribution Questions

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FAQs on Binomial Distribution

What is a binomial distribution.

The binomial distribution is a common discrete distribution used in statistics, as opposed to a continuous distribution, such as the normal distribution . The binomial distribution, therefore, represents the probability for x successes in n trials, given a success probability p for each trial, and is applicable to events having only two possible results in an experiment.

What Is the Purpose of Binomial Distribution?

The binomial distribution model allows us to compute the probability of observing a specified number of "successes" when the process is repeated a specific number of times (e.g., in a set of patients) and the outcome for a given patient is either a success or a failure.

What Is the Formula for Binomial Distribution?

The formula for binomial distribution is:

P(x: n,p) = n C x p x (q) n-x

n = the number of experiments
x = 0, 1, 2, 3, 4, …
p = Probability of success in a single experiment
q = Probability of failure in a single experiment (= 1 – p)

What Is the Formula for the Mean and Variance of the Binomial Distribution?

The mean and variance of the binomial distribution are:

Variance = npq

Where p is the probability of success, q is the probability of failure, and n = number of trials.

What Are the Criteria for the Binomial Distribution?

The criteria for using the binomial distribution are:

The number of trials should be fixed.
Each trial should be independent.
The probability of success is exactly the same from one trial to the other trial.

What Is the Difference Between a Binomial Distribution and Normal Distribution?

How do you identify a binomial distribution.

For a variable to be a binomial random variable, all of the following conditions must be met:

There are a fixed number of trials (a fixed sample size).
On each trial, the event of interest either occurs or does not.
The probability of occurrence (or not) is the same on each trial.
Trials are independent of one another.

Is Binomial Distribution Discrete or Continuous?

A binomial distribution is a discrete distribution with parameters n and p, where n is the number of trials and p is the probability of success.

For the Binomial Distribution, Which Formula Finds the Standard Deviation?

The standard deviation formula for a binomial distribution is given by, σ = √(npq), where n = number of trials, p = probability of success, q = probability of failure = 1 - p.

What is Negative Binomial Distribution?

Negative binomial distribution is a discrete probability distribution in statistics. It helps in finding r success in x trials. Here we consider the n + r trials needed to get r successes.

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Binomial Distribution in Probability

Binomial distribution is a fundamental probability distribution in statistics, used to model the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure. This distribution is particularly useful when you want to calculate the probability of a specific number of successes, such as flipping coins, quality control in manufacturing, or predicting survey outcomes.

Binomial Distribution in Probability can be implemented when there are two possible outcomes i.e. Success or Failure. In this article, we will learn about binomial probability distributions, its meaning, formulas and properties of a binomial distribution.

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What is Binomial Distribution in Probability?

Negative binomial distribution, binomial distribution formula, binomial random variable, binomial distribution calculation, binomial distribution examples, binomial distribution table, binomial distribution graph, binomial distribution in statistics, measure of central tendency for binomial distribution, binomial distribution mean, binomial distribution variance, binomial distribution standard deviation, binomial distribution properties, binomial distribution applications, binomial distribution vs normal distribution, binomial distribution in probability – solved examples, practice problems on binomial distribution in probability.

Binomial Probability Distribution talks about the probability of success or failure of an outcome in a series of events . The Binomial Distribution in Probability maps the outcome obtained in the form of success or failure, yes or no, true or false, etc. Each trial done to obtain the outcome of success or failure is called the Bernoulli Trial and the probability distribution for each Bernoulli Trial is called the Bernoulli Distribution. Let’s learn the definition and meaning of Binomial Distribution.

Binomial Distribution Definition

Binomial Distribution for a Random Variable X = 0, 1, 2, …., n is defined as the probability distribution of two outcomes success or failure in a series of events. Binomial Distribution in statistics uses one of the two independent variables in each trial where the outcome of each trial is independent of the outcome of other trials.

Condition for Binomial Distribution

The Binomial distribution can be used in scenarios where the following conditions are satisfied:

Fixed Number of Trials : There are a set number of trials or experiments (denoted by n), such as flipping a coin 10 times.
Two Possible Outcomes : Each trial has only two possible outcomes, often labeled as “success” and “failure.” For example, getting heads or tails in a coin flip.
Independent Trials : The outcome of each trial is independent of the others, meaning the result of one trial does not affect the result of another.
Constant Probability : The probability of success (denoted by p) remains the same for each trial. For example, if you’re flipping a fair coin, the probability of getting heads is always 0.5.

The Binomial distribution is an appropriate model to use for calculating the probabilities of obtaining a certain number of successes in the given trials.

Read More: Statistics in Maths

Binomial Distribution Meaning

The meaning of Binomial Distribution can be understood with the help of an example. Let us assume an event A. If we conduct a random experiment that gives us ‘A’ then we call it success ‘S’ and if not then we call it failure ‘F’. The probability of success is given as P(S) = p and the probability of failure is given as P(F) = 1. We know that in a random experiment either we will succeed in getting A or fail in getting A, i.e. we have only two sets of probability, either of failure or success. Hence, the Binomial Distribution gives the idea of the likelihood of an event for each trial where there are possibilities of only resulting in the event or not resulting in the event.

Consider a situation where getting 6 is the success on throwing a die. Now if we throw the die and not get 6 then it is a failure. Now we throw again and do not get 6. Let’s say we don’t get 6 for three successive attempts and six is obtained in the fourth attempt and onwards then the binomial distribution of the number of getting 6 is called the Negative Binomial Distribution.

Negative Binomial Distribution Formula

Formula for Negative Binomial Distribution is given as

P(x) = n+r-1 C r-1 p r q n

n is Total Number of Trial
r is Number of Trials in which we get the first success
p is Probability of Success in Each Trial
q is Probability of Failure in Each Trial

Binomial Distribution Formula which is used to calculate the probability, for a random variable X = 0, 1, 2, 3,….,n is given as

P(X = r) = n C r p r q n-r , r = 0, 1, 2, 3….

p is success
q is failure and q = 1 – p
p, q > 0 such that p + q = 1

Bernoulli Trials in Binomial Distribution

Bernoulli Trial is a trial that gives results of dichotomous nature i.e. result in yes or no, head or tail, even or odd. It means it gives two types of outcomes out of which one favors the event while the other doesn’t. A random experiment is called Bernoulli Trial if it satisfies the following conditions:

Trials are finite in number
Trials are independent of each other
Each trial has only two possible outcomes
Probability of success and failure in each trial is the same.

A Binomial Random Variable can be defined by two possible outcomes such as “success” and a “failure”. For instance, consider rolling a fair six-sided die and recording the value of the face. The binomial distribution formula can be put into use to calculate the probability of success for binomial distributions. Often it states “plugin” the numbers to the formula and calculates the requisite values.

Binomial distribution is based on the following characteristics:

Experiment contains n identical trials.
Each trial results in one of the two outcomes either success or failure.
Probability of success, denoted p, remains the same from trial to trial.
All the n trials are independent.

For Example, consider the following instance

A fair coin is flipped 20 times; X represents the number of heads.

X is a binomial random variable with n = 20 which is the total number of trials and p = 1/2 is the probability of getting head in each trial. The value of X represents the number of trials in which you succeed in getting head.

Binomial Distribution in statistics is used to compute the probability of likelihood of an event using the above formula. To calculate the probability using binomial distribution we need to follow the following steps:

Step 1: Find the number of trials and assign it as ‘n’ Step 2: Find the probability of success in each trial and assign it as ‘p’ Step 3: Find the probability of failure and assign it as q where q = 1-p Step 4: Find the random variable X = r for which we have to calculate the binomial distribution Step 5: Calculate the probability of Binomial Distribution for X = r using the Binomial Distribution Formula.

The use of the above steps has been illustrated using an example below:

Finding probability of getting exactly 6 heads when a fair coin is flipped 10 times .
Finding the probability of exactly 3 bulbs being defective when a batch of 100 bulbs is tested and each bulb has a 2% chance of being defective.
to find Probability of exactly 7 patients responding positively to the treatment when the drug is tested on 8 patients and has a 90% success rate.

Let’s say we toss a coin twice, and getting head is a success we have to calculate the probability of success and failure then, in this case, we will calculate the probability distribution as follows:

In each trial getting a head that is a success, its probability is given as p = 1/2

n = 2 as we throw a coin twice

r = 0 for no success, r = 1 for getting head once and r = 2 for getting head twice

Probability of failure q = 1 – p = 1 – 1/2 = 1/2.

P(Getting 1 head) = P(X = 1) = n c r p r q n-r = 2 c 1 (1/2) 1 (1/2) 1 = 2 ⨯ 1/2 ⨯ 1/2 = 1/2

P(Getting 2 heads) = P(X = 2) = 2 c 2 (1/2) 2 (1/2) 0 = 1/4

P(Getting 0 heads) = P(X = 0) = 2 c 0 (1/2) 0 (1/2) 2 = 1/4

Random Variable (X = r)	P(X = r)
X = 0 (Getting 0 Head)	1/4
X = 1 (Getting 1 Head)	1/2
X = 2 (Getting 2 Head)	1/4

As of now, we know that Binomial Distribution is calculated for the Random Variables obtained in Bernoulli Trials. Hence, we should understand these terms.

Binomial distribution for a situation when getting 6 is a success on throwing two dies is discussed in this section. First of all, we see that it is a Bernoulli Trial as getting 6 is the only success, and getting any different is a failure. Now we can get six on both die in a trial or six on only one of the die in a trial and getting no six on both die. Hence, the random variable for which we have to find the probability takes the value X = r = 0, 1, 2.

Binomial Distribution Table for getting 6 as success is plotted below:

Random Variable (X = r)	P(X = r)
X = 0 (Getting no 6)	25/36
X = 1 (Getting one 6)	10/36
X = 2 (Getting two 6)	1/36

We see that sum of all the probabilities 25/36 + 10/36 + 1/36 = 1.

Binomial Distribution Graph is plotted for X and P(X). We will plot a Binomial Distribution Graph for tossing a coin twice where getting the head is a success. If we toss a coin twice, the possible outcomes are {HH, HT, TH, TT}. Binomial Distribution Table for this is given below:

X (Random Variable)	P(X)
X = 0 (Getting no head)	P(X = 0) = 1/4 = 0.25
X = 1 (Getting 1 head)	P(X = 1) = 2/4 = 1/2 = 0.5
X = 2 (Getting two heads)	P(X = 2) = 1/4 = 0.25

Binomial Distribution Graph for the above table is given below:

Measures of central tendency, specifically the mean, provide insights into the distribution’s central or typical value for the number of successes in a series of independent trials. . For a binomial distribution defined by parameters n (number of trials) and p (probability of success on each trial), the measures of central tendency are characterized as follows:

In this, we will learn the formulas for Mean , Variance, and Standard Deviation of Binomial Distribution.

Mean of Binomial Distribution is the measurement of average success that would be obtained in ‘n’ number of trials. The Mean of Binomial Distribution is also called Binomial Distribution Expectation. The formula for Binomial Distribution Expectation is given as:

μ is Mean or Expectation
n is Total Number of Trials

Read more about, Expected Value or Expectation

Example: If we toss a coin 20 times and getting head is the success then what is the mean of success?

Total Number of Trials n = 20 Probability of getting head in each trial, p = 1/2 = 0.5 Mean = n.p = 20 ⨯ 0.5 It means on average we would head 10 times on tossing a coin 20 times.

Variance of Binomial Distribution tells about the dispersion or spread of the distribution. It is given by the product of the number of trials, probability of success, and probability of failure. The formula for Variance is given as follows:

σ 2 = n.p.q

σ 2 is Variance

Example: If we toss a coin 20 times and getting head is the success then what is the variance of the distribution?

We have, n = 20 Probability of Success in each trial (p) = 0.5 Probability of Failure in each trial (q) = 0.5 Variance of the Binomial Distribution, σ = n.p.q = (20 ⨯ 0.5 ⨯ 0.5) = 5

Standard Deviation of Binomial Distribution tells about the deviation of the data from the mean. Mathematically, Standard Deviation is the square root of the variance. The formula for the Standard Deviation of Binomial Distribution is given as

σ is Standard Deviation

Example: If we toss a coin 20 times and getting head is the success then what is the standard deviation?

We have, n = 20 Probability of Success in each trial (p) = 0.5 Probability of Failure in each trial (q) = 0.5 Standard Deviation of the Binomial Distribution, σ = √n.p.q ⇒ σ = √(20 ⨯ 0.5 ⨯ 0.5) ⇒ σ = √5 = 2.23

Properties of Binomial Distribution are mentioned below:

There are only two possible outcomes: success or failure, yes or no, true or false.
There is a finite number of trials given as ‘n’.
Only Success is calculated out of all trials.
Each trial is independent of any other trial.

Binomial Distribution is used where we have only two possible outcomes. Let’s see some of the areas where Binomial Distribution can be used

To find the number of male and female students in an institute.
To find the likeability of something in Yes or No.
To find defective or good products manufactured in a factor.
To find positive and negative reviews on a product.
Votes collected in the form of 0 or 1.

Binomial Distribution differs from the Normal Distribution in many aspects. The key differences and characteristics of the Binomial and Normal distributions are highlighted in the following table:


	Discrete probability distribution	Continuous probability distribution
	Two possible outcomes per trial (success or failure)	Infinite possible outcomes within a continuous range
	n (number of trials), p (probability of success)	μ (mean), σ (standard deviation)
	Varies depending on n and p; typically skewed unless p=0.5 and n is large	Bell-shaped curve (symmetric)
	xxx can take integer values from 0 to n	x can take any real number (from −∞ to +∞)
	μ=np	μ
	[Tex]\sigma^2 = np(1-p)[/Tex]	[Tex]\sigma^2[/Tex]
	Used for modeling the number of successes in a fixed number of independent trials	Used for modeling continuous data that clusters around a mean
	Flipping coins, quality control (defective items)	Heights of people, test scores, measurement errors
	Approximates Normal distribution for large n and p not too close to 0 or 1	Considered the limit of the Binomial distribution as n becomes large and p is near 0.5

Binomial Distribution in Probability – FAQs

What is binomial distribution in maths.

Binomial Distribution is the probability distribution of the success of obtained in a Bernoulli Trial.

What is Binomial Distribution Formula?

The Binary Distribution Formula is given as P(X = r) = n C r p r q n-r . Here r = 0, 1, 2, 3 … Where, p is success, q is failure and is given by q = 1 – p, and p, q > 0 such that p + q = 1.

What are Binomial Distribution Conditions?

Conditions for Binomial Distribution are mentioned below, Only two possible outcomes such as success or failure, yes or no, true or false. There is a finite number of trials given as ‘n’. Probability of success and failure in each trial is the same. Only Success is calculated out of all trial. Each trial is independent of any other trial.

What is Binomial Distribution Mean and Variance?

Binomial Distribution Mean tells about the average success obtained in ‘n’ number of trials. Binomial Distribution Mean is also called Binomial Distribution Expectation. The formula for Binomial Distribution Expectation is given as μ = n.p. Binomial Distribution Variance is the measurement of the spread of the distribution. The formula of the variance is given by σ 2 = n.p.q.

What are the Binomial Distribution Characteristics?

Characteristics of Binomial Distribution are mentioned as follows: Possible outcomes such as success or failure, yes or no, true or false. Number of observations is finite ‘n’. Probability of success and failure in each trial is the same. Each trial is independent of any other trials.

What Is the Purpose of Binomial Distribution Formula?

Purpose of Binomial Distribution formula is to calculate the probability of obtaining a specific number of successes (often denoted as “k”) in a fixed number of independent and identical trials (often denoted as “n”) when each trial has only two possible outcomes: success or failure.

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Binomial Experiments: An Explanation + Examples
This tutorial defines a binomial experiment and provides several examples of experiments that are and are not considered to be binomial experiments. Binomial Experiment: Definition. A binomial experiment is an experiment that has the following four properties: 1. The experiment consists of n repeated trials. The number n can be any amount.
Binomial Experiment: Rules, Examples, Steps
Step 1: Ask yourself: is there a fixed number of trials? For question #1, the answer is yes (200). For question #2, the answer is no, so we're going to discard #2 as a binomial experiment. For question #3, the answer is yes, there's a fixed number of trials (the 50 traffic lights). For question #4, the answer is yes (your 6 darts).
Binomial Experiments: An Explanation + Examples
This tutorial defines a binomial experiment and provides several examples of experiments that are and are not considered to be binomial experiments. Binomial Experiment: Definition. A binomial experiment is an experiment that has the following four properties: 1. The experiment consists of n repeated trials. The number n can be any amount.
An Introduction to the Binomial Distribution
The binomial distribution is one of the most popular distributions in statistics.To understand the binomial distribution, it helps to first understand binomial experiments.. Binomial Experiments. A binomial experiment is an experiment that has the following properties:. The experiment consists of n repeated trials.; Each trial has only two possible outcomes.
Binomial distribution
In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes-no question, and each with its own Boolean-valued outcome: success (with probability p) or failure (with probability q = 1-p).
4.3 Binomial Distribution
The outcomes of a binomial experiment fit a binomial probability distribution. The random variable X = the number of successes obtained in the n independent trials. The mean, μ, and variance, σ 2, for the binomial probability distribution are μ = np and σ 2 = npq. The standard deviation, σ, is then σ = n p q n p q.
Binomial Distribution
The probability of success is the same every time the experiment is repeated. A binomial experiment is a series of $n$ Bernoulli trials, whose outcomes are independent of each other. A random variable, $X$, is defined as the number of successes in a binomial experiment. Finally, a binomial distribution is the probability distribution of $X$.
Binomial distribution
The binomial distribution is characterized as follows. Definition Let be a discrete random variable. Let and . Let the support of be We say that has a binomial distribution with parameters and if its probability mass function is where is a binomial coefficient. The following is a proof that is a legitimate probability mass function.
4.3 Binomial Distribution
Expected Value and Standard Deviation. The mean, μ μ, and variance, σ2 σ 2, for the binomial probability distribution are μ = np μ = n p and σ2 = npq σ 2 = n p q. The standard deviation is then σ = √npq σ = n p q. Example. In the 2013 Jerry's Artarama art supplies catalog, there are 560 pages. Eight of the pages feature signature ...
5.2: Binomial Probability Distribution
Properties of a binomial experiment (or Bernoulli trial) Homework; Section 5.1 introduced the concept of a probability distribution. The focus of the section was on discrete probability distributions (pdf). To find the pdf for a situation, you usually needed to actually conduct the experiment and collect data.
11.2: The Binomial Distribution
Compare the Poisson experiment and the binomial timeline experiment. Open the Poisson experiment and set $ r = 1 $ and $ t = 5 $. Run the experiment a few times and note the general behavior of the random points in time. Note also the shape and location of the probability density function and the mean$ \pm $standard deviation bar.
7.1: Binomial Experiments and Distributions
Definition; binomial experiments: Binomial experiments are experiments that include only two choices, with distributions that involve a discrete number of trials of these two possible outcomes. binomial random variable: A binomial random variable is a type of random variable that can only be used to count whether a certain event occurs or does ...
What are Binomial Experiments?
There are three characteristics of a binomial experiment. There are a fixed number of trials. Think of trials as repetitions of an experiment. The letter n n denotes the number of trials. There are only two possible outcomes, called "success" and "failure," for each trial. The letter p p denotes the probability of a success on one trial ...
Binomial Experiment
Definition. A binomial experiment is a statistical experiment that meets the following criteria: 1) the experiment consists of a fixed number of trials, 2) each trial has only two possible outcomes, often referred to as 'success' and 'failure', 3) the probability of success is the same for each trial, and 4) the trials are independent of one another.
5.3: Binomial Distribution
The outcomes of a binomial experiment fit a binomial probability distribution. The random variable $X =$ the number of successes obtained in the $n$ independent trials. The mean, $\mu$, and variance, $\sigma^{2}$, for the binomial probability distribution are ... In a statistics class of 50 students, what is the probability that at ...
Binomial Probability Distribution
Cumulative Binomial Probability. A cumulative binomial probability refers to the probability that the binomial random variable falls within a specified range (e.g., is greater than or equal to a stated lower limit and less than or equal to a stated upper limit).. To compute a cumulative binomial probability, we find the sum of relevant individual binomial probabilities, as illustrated in the ...
Binomial Distribution
In probability theory and statistics, the binomial distribution is the discrete probability distribution that gives only two possible results in an experiment, either Success or Failure.For example, if we toss a coin, there could be only two possible outcomes: heads or tails, and if any test is taken, then there could be only two results: pass or fail.
Binomial Theorem: Simple Definition, Formula, Step by Step Videos
Example. If you were to roll a die 20 times, the probability of you rolling a six is 1/6. This ends in a binomial distribution of (n = 20, p = 1/6). For rolling an even number, it's (n = 20, p = ½). Dice rolling is binomial. There are hundreds of ways you could measure success, but this is one of the simplest. Something works, or it doesn't.
Binomial Distribution Formula: Probability ...
For our die example we have n = 10 rolls, a success probability of p = 0.1667, and a failure probability of (1 - p) = 0.833. Let's enter these values into the formula. 10 * 0.1667 * 0.8333 = 1.3891. That's the variance, which uses squared units. To find the standard deviation of the binomial distribution, we need to take the square root ...
Binomial Distribution: Definition, Formula, Analysis, and Example
Binomial distribution is used to figure the probability of a pass or fail outcome in a survey, or experiment replicated numerous times. There are only two potential outcomes for this type of ...
Introduction to Binomial Distributions
Revision notes on Introduction to Binomial Distributions for the College Board AP® Statistics syllabus, written by the Statistics experts at Save My Exams. ... Probability & Statistics 1; Probability & Statistics 2; Practice Papers; Edexcel IAL. Maths. Pure 1; Pure 2; Pure 3; Pure 4; Mechanics 1; Mechanics 2; Statistics 1; Statistics 2 ...
Binomial Distribution
Binomial Distribution. In statistics and probability theory, the binomial distribution is the probability distribution that is discrete and applicable to events having only two possible results in an experiment, either success or failure. (the prefix "bi" means two, or twice). A few circumstances where we have binomial experiments are tossing a coin: head or tail, the result of a test ...
Binomial Distribution in Probability
Binomial distribution is a fundamental probability distribution in statistics, used to model the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure. This distribution is particularly useful when you want to calculate the probability of a specific number of successes, such as flipping coins, quality control in ...
Khan Academy
Learn how to calculate the probability of a binomial random variable with examples and exercises. Khan Academy offers free, world-class education for anyone, anywhere.

Binomial Experiments: An Explanation + Examples

Binomial Experiment: Definition

Examples of Binomial Experiments

Examples that are not Binomial Experiments

A Binomial Experiment Example & Solution

How to Create a Residual Plot in Excel

4.3 Binomial Distribution

Example 4.9

Example 4.10

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Example 4.11

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Example 4.12

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Notation for the Binomial: B = Binomial Probability Distribution Function

Example 4.13

Using the TI-83, 83+, 84, 84+ Calculator

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Example 4.14

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Example 4.15

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Example 4.16

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Binomial Distribution

Formal Definition

This problem is part of the set Extremely Biased Coins.

Binomial distribution

How the distribution is used

Chart of binomial distribution with interactive calculator

How to cite

4.3 Binomial Distribution

Binomial Experiments

Notation for the Binomial

Experiments That Are Not Binomial

Binomial Probability Function

Expected Value and Standard Deviation

Media Attributions

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Module 4: Discrete Random Variables

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5 Must Know Facts For Your Next Test

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Binomial Probability Distribution

Binomial Experiment

Binomial Distribution

Binomial Formula and Binomial Probability

Cumulative Binomial Probability

Binomial Calculator

Binomial Distribution

Negative Binomial Distribution

Binomial Distribution Vs Normal Distribution

Practice Problems

Binomial Distribution Examples

Binomial Distribution Formula

Binomial Distribution Mean and Variance

Properties of Binomial Distribution

Binomial Distribution Examples And Solutions

Frequently Asked Questions on Binomial Distribution

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Binomial Theorem: Simple Definition, Formula, Step by Step Videos

Basic Information on the Binomial Theorem

Binomial Theorem: More Advanced Topics

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Binomial Expansions and Series

Binomial Distribution Formula: Probability, Standard Deviation & Mean

Binomial Distribution Formula

Worked Example of Finding a Binomial Probability

Expected Value of Binomial Distribution

Standard Deviation of the Binomial Distribution

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