theoretical probability. experimental probability. axiomatic probability

What is Probability and Different Types of Probability

Types of probability are one of the most important topics of mathematics. It is present in the curriculum of lower as well as higher classes. Because it helps us in many ways, like from solving mathematics problems to a real-life situation. 

The probability is everywhere.  Let’s take a few examples of it.

• Planning about the weather

Meteorologists use various instruments to predict whether it will rain or not on a particular day. 

For instance, if it is said that there is a 60% chance of rain, it means that 60 out of 100 days would be the chance of rain. Therefore, it would be better to wear rain shoes instead of sandals. 

• Sports Strategies

Coaches and Athletes use probability concepts to examine the best sports strategies for the competitions and games. A cricket coach estimates the batting average of the player when he lines up the players. 

For instance, the player that has a 200 batting average signifies that the player hit 2 out of each 10 at the bat. At the same time, the player with a 400 batting average has more tendency to hit the ball- 4 out of every 10 balls.

What Is Probability?

Table of Contents

So you might have a little bit of idea about probability really is, but keeping that aside as we will discuss what probability means.

Probability = Possibility; in the short term, the possibility of getting something done or the possibility of solving some problem or the possibility of doing something. And also, there are different types of probability which we will be discussing below. 

Terms Related To Probability

There are various terms for probability, here we will discuss few of them : 

An event refers to an outcome or set of random experiment outcomes. It can be a single outcome or a combination of outcomes.

2. Sample Space

The sample space is the set of all possible outcomes of a random experiment. It represents the complete set of events that could potentially occur.

3. Experiment

An experiment is a process or an activity that results in an outcome. In the context of probability, it refers to a situation where the outcome is uncertain or random.

4. Independent Events

If one event happens or does not happen, it does change the chance that the other will happen. Independent events are events where the occurrence or non-occurrence of one event does not affect the probability of the other event. The outcomes of independent events are statistically unrelated.

5. Dependent Events

Dependent events are ones whose chances of happening depend on how often another event happens. The outcomes of dependent events are statistically related.

6. Expected Value

The expected value is the average of a random variable’s values, weighted by how likely each value is. It represents the long-term average outcome of an experiment.

Applications Of Probability

Here are some applications of probability explained in simple language:

1. Risk Assessment

Probability is essential in assessing risks in various fields. Insurance companies use probability to calculate premiums by considering the likelihood of an event occurring and its potential costs. Similarly, in finance, probability is used to evaluate investment risks and make informed decisions.

2. Medical Diagnoses

In healthcare, probability is used to evaluate the likelihood of diseases or conditions based on symptoms, test results, and patient history. Doctors use probabilistic reasoning to diagnose and determine the most appropriate treatment options.

3. Quality Control

Probability is used in manufacturing and quality control processes. It helps determine the probability of defects or errors occurring during production and allows companies to identify and address potential issues before they become significant problems.

4. Genetics and Biology

Probability is used in genetics to study the likelihood of specific traits or diseases being passed down from parents to offspring. It is also used in biological experiments to analyze the probability of certain outcomes or events occurring.

5. Traffic Planning

Probability is used in traffic planning and engineering to estimate the flow of vehicles and analyze the likelihood of traffic congestion at different times and locations. This information helps in designing efficient road systems and managing traffic effectively.

Why Probability Is Important?

Here are five points explaining why probability is important:

• Probability allows us to assess and manage risks effectively, guiding decision-making processes and enabling us to allocate resources wisely.
• It forms the foundation of statistical analysis, enabling researchers to draw valid conclusions from data, make predictions, and understand complex phenomena.
• Probability provides a framework for making informed decisions in various fields, including finance, healthcare, and engineering, by quantifying uncertainties and evaluating potential outcomes.
• It plays a crucial role in scientific research, helping scientists design experiments, analyze data, and draw reliable conclusions about the natural world.
• Probability is essential in predictive modeling, enabling the estimation of future events and outcomes, leading to improved planning, forecasting, and decision-making in diverse domains.

What Is The Value Of Probability?

As we have discussed above, it is one of the most important mathematics branches, and it deals with the occurrence of random events. Probability helps understand some events that occur or not or the percentage of occurrence of that particular event. 

The value of probability for occurring of a random event is always expressed between 0 and 1, so basically, from all this above information, we can say that the probability was introduced in mathematics for getting to know about the occurrence of some events. Or we can say that it helps us predict how likely events will happen.

Some Key Points About Probability

There is a basic theory associated with branch probability of random method. The meaning of probability is the chances of something likely to happen. This is the same thing as above, and that is the possibility of occurrence of an event. And all and all, this is also the probability theory used in the theory of probability distribution.

In the probability distribution theory , you will check that the probability of some outcome from any random experiment is based on the probability of any single element occurring from the number of Total possible events.

You can also say that to find the probability of any given situation or entire population. We need to know about the total possible outcomes of that situation. Only then can we know about the probability of a single event occurring from those situations.

And one of the most important things in Probability is the probability of all the events. That is happening in any situation sums up to 1.

This is one of the most important things to know or to remember whenever you are working on a probability problem or a real-life situation that involves probability of  well-defined data to get it solved.

For example, whenever we are going to Toss a Coin, there can be only two outcomes: head (H) or tail (T). There is no chance that both of these outcomes come at one time.

But when we toss 2 coins together in their three possibilities can occur like both the coins can be heads, or both the coins show tails or from both of those coins either one can be head, and another can be tail. That is (H H), (T T), (H T), and (T H). This is how we will get to know about a single event’s probability from the series of events.

What Is The Formula Of Probability?

Now, it is easy to use the formula of probability everywhere, such as to know the birth probability of a larger population. That is defined as the possibility of the occurring element being equal to the ratio of a number of favorable outcomes and the number of Total outcomes. 

This means the probability of an event P(E) of a sample size is equal to the number of favorable outcomes divided by the total number of that situation’s outcome.

P(E)= number of favorable outcomes / total number of outcome 

Example: There are 10 pillows in a bed; 2 are blue, 5 are yellow, and 3 are red. Calculate the probability of selecting a blue pillow?

The probability of selecting the blue pillow is equal to the number of blue pillows divided by the total number of pillows, 

=> 2/10 = 1/5 = 0.20

What Are The Different Types Of Probability?

Now, as we have already discussed, what probability is and its basic formula of the probability. Now it’s time to discuss the types of probability; you read it right. There are three major types of probabilities, and those are:

• Theoretical probability
• Experimental probability
• Axiomatic probability

1. Theoretical probability

Theoretical probability is based on the chances of something happening. We can also say that it is based on the possible chances of things happening in a particular problem, or previous events or a real-life situation. The probability is basically based on the basic reasoning open probability.

2. Experimental Probability

The name suggests that it is experimental. It means it will consist of some experiments in this type of probability. Basically, we can say that the experimental probability is based on the observation coming from an experiment.

In order to get an answer from such a type of probability, there must be an experiment going on, and from that, we will account or observe the outcomes, and then we will get to know about the probability of any event from that particular experiment.

Note: The experimental probability can be counted as the number of possible outcomes by the number of trials because we are experimenting, and experiments are based on different trials. So the experimental probability will be equal to two possible outcomes by the total number of trials.

3. Axiomatic probability

There is a set of rules in axiomatic probability, or we can call those sets of rules axioms. These rules get applied to all the types of reasons for a set of rules known as Kolmogorov’s three axioms. With the help of axiomatic probability, we can calculate the chances of occurrence and non-occurrence of any event.

And the axiomatic perspective says that probability is any function (we can call it P) from events to numbers satisfying the three conditions (axioms).

And those three conditions are:- 

Formula For The Probability Of An Event

You already know about what probability is and what are the types of probability now you should know about one of the most important formulas. Which is used many times in the branch of probability and regardless of the types of probability this formula is used everywhere. 

P(E’) = n-r/n = 1-r/n = 1-P(E)

So, P(E) + P(E’) = 1

This means that the total number or sum of probability can never be more than one. 

What Are Equally Likely Events?

When the same theoretical probability of happening, then the probability is known as equally likely events. A sample space results are called equally likely if each event has a similar probability of occurring. 

For instance, if a person throws a die, then the probability of occurring 1 is 1/6. Similarly, the probability of occurring all other numbers from 2 to 6, one at the same time, is 1/6.

Let’s test your knowledge about probability!

1. What is the probability that all the events add up in a given sample space?

2. A die is thrown. Calculate the probability of obtaining an odd number?

3. The probability that depends on the experiments’ observations is called:

• Theoretical Probability
• Experimental Probability
• Axiomatic Probability
• None of these

4. The complement of P(E) or P(E’) is:

5. If the numbers of events have the equivalent theoretical probability of occurring, then all are known as:

• Equally likely events
• Mutually exhaustive events
• Mutually exclusive events
• Impossible events

So, this was all about probability and the different types of probability. We hope that by reading this blog you will get all the essential knowledge needed for working in a branch of probability. If you are a student then this blog must help you with providing important knowledge about probability. So that you can get the most out whenever you study probability and the different types of probability. if you need the assignment of probability , then Get the best probability assignment help from our experts.

Frequently Asked Question

What is the difference between theoretical and experimental probability.

Theoretical probability is about what is supposed to occur or happen. Experimental probability is about the outcome of an experiment.

How many types of events are there in probability?

Types of Events within the probability:

Simple Events. Impossible and Sure Events. Dependent and Independent Events. Exhaustive Events. Events Associated with “OR” Compound Events. Mutually Exclusive Events. Complementary Events.

What is an example of an impossible event?

An impossible event is an event, which has the probability of zero and can not happen. E is an impossible event when & only when P(E) = 0. For example, flipping a coin once, there is an impossible event probability of getting BOTH a tail AND a head.

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Theoretical vs. Experimental Probability: How do they differ?

Probability is the study of chances and is an important topic in mathematics. There are two types of probability: theoretical and experimental.

So, how to define theoretical and experimental probability? Theoretical probability is calculated using mathematical formulas, while experimental probability is based on results from experiments or surveys. In order words, theoretical probability represents how likely an event is to happen. On the other hand, experimental probability illustrates how frequently an event occurs in an experiment.

Read on to find out the differences between theoretical and experimental probability. If you wonder How to Understand Statistics Easily , I wrote a whole article where I share 9 helpful tips to help you Ace statistics.

Table of Contents

What Is Theoretical Probability?

Theoretical probability is calculated using mathematical formulas. In other words, a theoretical probability is a probability that is determined based on reasoning. It does not require any experiments to be conducted. Theoretical probability can be used to calculate the likelihood of an event occurring before it happens.

Keep in mind that theoretical probability doesn’t involve any experiments or surveys; instead, it relies on known information to calculate the chances of something happening.

For example, if you wanted to calculate the probability of flipping a coin and getting tails, you would use the formula for theoretical probability. You know that there are two possible outcomes—heads or tails—and that each outcome is equally likely, so you would calculate the probability as follows: 1/2, or 50%.

How Do You Calculate Theoretical Probability?

• First, start by counting the number of possible outcomes of the event.
• Second, count the number of desirable (favorable) outcomes of the event.
• Third, divide the number of desirable (favorable) outcomes by the number of possible outcomes.
• Finally, express this probability as a decimal or percentage.

The theoretical probability formula is defined as follows: Theoretical Probability = Number of favorable (desirable) outcomes divided by the Number of possible outcomes.

How Is Theoretical Probability Used in Real Life?

Probability plays a vital role in the day to day life. Here is how theoretical probability is used in real life: 

• Sports and gaming strategies
• Analyzing political strategies.
• Buying or selling insurance
• Determining blood groups 
• Online shopping
• Weather forecast
• Online games

What Is Experimental Probability?

Experimental probability, on the other hand, is based on results from experiments or surveys. It is the ratio of the number of successful trials divided by the total number of trials conducted. Experimental probability can be used to calculate the likelihood of an event occurring after it happens.

For example, if you flipped a coin 20 times and got heads eight times, the experimental probability of obtaining heads would be 8/20, which is the same as 2/5, 0.4, or 40%.

How Do You Calculate Experimental Probability?

The formula for the experimental probability is as follows:  Probability of an Event P(E) = Number of times an event happens divided by the Total Number of trials .

If you are interested in learning how to calculate experimental probability, I encourage you to watch the video below.

How Is Experimental Probability Used in Real Life?

Knowing experimental probability in real life provides powerful insights into probability’s nature. Here are a few examples of how experimental probability is used in real life:

• Rolling dice
• Selecting playing cards from a deck
• Drawing marbles from a hat
• Tossing coins

The main difference between theoretical and experimental probability is that theoretical probability expresses how likely an event is to occur, while experimental probability characterizes how frequently an event occurs in an experiment.

In general, the theoretical probability is more reliable than experimental because it doesn’t rely on a limited sample size; however, experimental probability can still give you a good idea of the chances of something happening.

The reason is that the theoretical probability of an event will invariably be the same, whereas the experimental probability is typically affected by chance; therefore, it can be different for different experiments.

Also, generally, the more trials you carry out, the more times you flip a coin, and the closer the experimental probability is likely to be to its theoretical probability.

Also, note that theoretical probability is calculated using mathematical formulas, while experimental probability is found by conducting experiments.

What to read next:

• Types of Statistics in Mathematics And Their Applications .
• Is Statistics Harder Than Algebra? (Let’s find out!)
• Should You Take Statistics or Calculus in High School?
• Is Statistics Hard in High School? (Yes, here’s why!)

Wrapping Up

Theoretical and experimental probabilities are two ways of calculating the likelihood of an event occurring. Theoretical probability uses mathematical formulas, while experimental probability uses data from experiments. Both types of probability are useful in different situations.

I believe that both theoretical and experimental probabilities are important in mathematics. Theoretical probability uses mathematical formulas to calculate chances, while experimental probability relies on results from experiments or surveys.

I am Altiné. I am the guy behind mathodics.com. When I am not teaching math, you can find me reading, running, biking, or doing anything that allows me to enjoy nature's beauty. I hope you find what you are looking for while visiting mathodics.com.

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Introduction       types of mistakes        suggestions        resources      table of contents     about      glossary      blog, what is probability, four perspectives on probability, 1. classical (sometimes called "a priori" or "theoretical"), 2. empirical (sometimes called "a posteriori" or "frequentist"), 3. subjective, 4. axiomatic.

• 0 ≤ P(E) ≤ 1 for every allowable event E. (In other words, 0 is the smallest allowable probability and 1 is the largest allowa ble probability).
• The certain event has probability 1. (The certain event is the event "some outcome occurs." For example, in rolling a die, the certain event is "One of 1, 2, 3, 4, 5, 6 comes up." In considering the stock market, the certain event is "The Dow Jones either goes up or goes down or stays the same.")
• The probability of the union of mutually exclusive events is the sum of the probabilities of the individual events. (Two events are called mutually exclusive if they cannot both occur simultaneously. For example, the events "the die comes up 1" and "the die comes up 4" are mutually exclusive, assuming we are talking about the same toss of the same die. The union of events is the event that at least one of the events occurs. For example, if E is the event "a 1 comes up on the die" and F is the event "an even number comes up on the die," then the union of E and F is the event "the number that comes up on the die is either 1 or even."

Probability

Probability defines the likelihood of occurrence of an event. There are many real-life situations in which we may have to predict the outcome of an event. We may be sure or not sure of the results of an event. In such cases, we say that there is a probability of this event to occur or not occur. Probability generally has great applications in games, in business to make predictions, and also it has extensive applications in this new area of artificial intelligence.

The probability of an event can be calculated by the probability formula by simply dividing the favourable number of outcomes by the total number of possible outcomes. The value of the probability of an event happening can lie between 0 and 1 because the favourable number of outcomes can never be more than the total number of outcomes. Also, the favorable number of outcomes cannot be negative. Let us discuss the basics of probability in detail in the following sections.

What is Probability?

Probability can be defined as the ratio of the number of favorable outcomes to the total number of outcomes of an event. For an experiment having 'n' number of outcomes, the number of favorable outcomes can be denoted by x. The formula to calculate the probability of an event is as follows.

Probability(Event) = Favorable Outcomes/Total Outcomes = x/n

Probability is used to predict the outcomes for the tossing of coins, rolling of dice, or drawing a card from a pack of playing cards. The probability is classified into two types:

• Theoretical probability
• Experimental probability

To understand each of these types, click on the respective links.

Terminology of Probability Theory

The following terms in probability theorey help in a better understanding of the concepts of probability.

Experiment: A trial or an operation conducted to produce an outcome is called an experiment.

Sample Space: All the possible outcomes of an experiment together constitute a sample space . For example, the sample space of tossing a coin is {head, tail}.

Favorable Outcome: An event that has produced the desired result or expected event is called a favorable outcome. For example, when we roll two dice, the possible/favorable outcomes of getting the sum of numbers on the two dice as 4 are (1,3), (2,2), and (3,1).

Trial: A trial denotes doing a random experiment.

Random Experiment: An experiment that has a well-defined set of outcomes is called a random experiment . For example, when we toss a coin, we know that we would get ahead or tail, but we are not sure which one will appear.

Event: The total number of outcomes of a random experiment is called an event .

Equally Likely Events: Events that have the same chances or probability of occurring are called equally likely events. The outcome of one event is independent of the other. For example, when we toss a coin, there are equal chances of getting a head or a tail.

Exhaustive Events: When the set of all outcomes of an event is equal to the sample space, we call it an exhaustive event .

Mutually Exclusive Events: Events that cannot happen simultaneously are called mutually exclusive events . For example, the climate can be either hot or cold. We cannot experience the same weather simultaneously.

Events in Probability

In probability theory, an event is a set of outcomes of an experiment or a subset of the sample space. If P(E) represents the probability of an event E, then, we have,

• P(E) = 0 if and only if E is an impossible event.
• P(E) = 1 if and only if E is a certain event.
• 0 ≤ P(E) ≤ 1.

Suppose, we are given two events, "A" and "B", then the probability of event A, P(A) > P(B) if and only if event "A" is more likely to occur than the event "B". Sample space(S) is the set of all of the possible outcomes of an experiment and n(S) represents the number of outcomes in the sample space.

P(E) = n(E)/n(S)

P(E’) = (n(S) - n(E))/n(S) = 1 - (n(E)/n(S))

E’ represents that the event will not occur.

Therefore, now we can also conclude that, P(E) + P(E’) = 1

Probability Formula

The probability equation defines the likelihood of the happening of an event. It is the ratio of favorable outcomes to the total favorable outcomes. The probability formula can be expressed as,

i.e., P(A) = n(A)/n(S)

• P(A) is the probability of an event 'B'.
• n(A) is the number of favorable outcomes of an event 'B'.
• n(S) is the total number of events occurring in a sample space.

Different Probability Formulas

Probability formula with addition rule : Whenever an event is the union of two other events, say A and B, then P(A or B) = P(A) + P(B) - P(A∩B) P(A ∪ B) = P(A) + P(B) - P(A∩B)

Probability formula with the complementary rule: Whenever an event is the complement of another event, specifically, if A is an event, then P(not A) = 1 - P(A) or P(A') = 1 - P(A). P(A) + P(A′) = 1.

Probability formula with the conditional rule : When event A is already known to have occurred, the probability of event B is known as conditional probability and is given by: P(B∣A) = P(A∩B)/P(A)

Probability formula with multiplication rule : Whenever an event is the intersection of two other events, that is, events A and B need to occur simultaneously. Then

• P(A ∩ B) = P(A)⋅P(B) (in case of independent events )
• P(A∩B) = P(A)⋅P(B∣A) (in case of dependent events )

Calculating Probability

In an experiment, the probability of an event is the possibility of that event occurring. The probability of any event is a value between (and including) "0" and "1". Follow the steps below for calculating probability of an event A:

• Step 1: Find the sample space of the experiment and count the elements. Denote it by n(S).
• Step 2: Find the number of favorable outcomes and denote it by n(A).
• Step 3: To find probability, divide n(A) by n(S). i.e., P(A) = n(A)/n(S).

Here are some examples that well describe the process of finding probability.

Example 1 : Find the probability of getting a number less than 5 when a dice is rolled by using the probability formula.

To find: Probability of getting a number less than 5 Given: Sample space, S = {1,2,3,4,5,6} Therefore, n(S) = 6

Let A be the event of getting a number less than 5. Then A = {1,2,3,4} So, n(A) = 4

Using the probability equation, P(A) = (n(A))/(n(s)) p(A) = 4/6 m = 2/3

Answer: The probability of getting a number less than 5 is 2/3.

Example 2: What is the probability of getting a sum of 9 when two dice are thrown?

There is a total of 36 possibilities when we throw two dice. To get the desired outcome i.e., 9, we can have the following favorable outcomes. (4,5),(5,4),(6,3)(3,6). There are 4 favorable outcomes. Probability of an event P(E) = (Number of favorable outcomes) ÷ (Total outcomes in a sample space) Probability of getting number 9 = 4 ÷ 36 = 1/9

Answer: Therefore the probability of getting a sum of 9 is 1/9.

Probability Tree Diagram

A tree diagram in probability is a visual representation that helps in finding the possible outcomes or the probability of any event occurring or not occurring. The tree diagram for the toss of a coin given below helps in understanding the possible outcomes when a coin is tossed. Each branch of the tree is associated with the respective probability (just like how 0.5 is written on each brack in the figure below). Remember that the sum of probabilities of all branches that start from the same point is always 1 (here, 0.5 + 0.5 = 1).

Types of Probability

There can be different perspectives or types of probabilities based on the nature of the outcome or the approach followed while finding probability of an event happening. The four types of probabilities are,

Classical Probability

Empirical probability, subjective probability, axiomatic probability.

Classical probability, often referred to as the "priori" or "theoretical probability", states that in an experiment where there are B equally likely outcomes, and event X has exactly A of these outcomes, then the probability of X is A/B, or P(X) = A/B. For example, when a fair die is rolled, there are six possible outcomes that are equally likely. That means, there is a 1/6 probability of rolling each number on the die.

The empirical probability or the experimental perspective evaluates probability through thought experiments. For example, if a weighted die is rolled, such that we don't know which side has the weight, then we can get an idea for the probability of each outcome by rolling the die number of times and calculating the proportion of times the die gives that outcome and thus find the probability of that outcome.

Subjective probability considers an individual's own belief of an event occurring. For example, the probability of a particular team winning a football match on a fan's opinion is more dependent upon their own belief and feeling and not on a formal mathematical calculation.

In axiomatic probability, a set of rules or axioms by Kolmogorov are applied to all the types. The chances of occurrence or non-occurrence of any event can be quantified by the applications of these axioms, given as,

• The smallest possible probability is zero, and the largest is one.
• An event that is certain has a probability equal to one.
• Any two mutually exclusive events cannot occur simultaneously, while the union of events says only one of them can occur.

Coin Toss Probability

Let us now look into the probability of tossing a coin . Quite often in games like cricket, for making a decision as to who would bowl or bat first, we sometimes use the tossing of a coin and decide based on the outcome of the toss. Let us check how we can use the concept of probability in the tossing of a single coin. Further, we shall also look into the tossing of two and three coins.

Tossing a Coin

A single coin on tossing has two outcomes, a head, and a tail. The concept of probability which is the ratio of favorable outcomes to the total number of outcomes can be used in finding probability of getting the head and the probability of getting a tail.

Total number of possible outcomes = 2; Sample Space = {H, T}; H: Head, T: Tail

• P(H) = Number of heads/Total outcomes = 1/2
• P(T)= Number of Tails/ Total outcomes = 1/2

Tossing Two Coins

In the process of tossing two coins, we have a total of four (= 2 2 ) outcomes. The probability formula can be used to find the probability of two heads, one head, no head, and a similar probability can be calculated for the number of tails. The probability calculations for the two heads are as follows.

Total number of outcomes = 4; Sample Space = {(H, H), (H, T), (T, H), (T, T)}

• P(2H) = P(0 T) = Number of outcome with two heads/Total Outcomes = 1/4
• P(1H) = P(1T) = Number of outcomes with only one head/Total Outcomes = 2/4 = 1/2
• P(0H) = (2T) = Number of outcome with two heads/Total Outcomes = 1/4

Tossing Three Coins

The number of total outcomes on tossing three coins simultaneously is equal to 2 3 = 8. For these outcomes, we can find the probability of getting one head, two heads, three heads, and no head. A similar probability can also be calculated for the number of tails.

Total number of outcomes = 2 3 = 8 Sample Space = {(H, H, H), (H, H, T), (H, T, H), (T, H, H), (T, T, H), (T, H, T), (H, T, T), (T, T, T)}

• P(0H) = P(3T) = Number of outcomes with no heads/Total Outcomes = 1/8
• P(1H) = P(2T) = Number of Outcomes with one head/Total Outcomes = 3/8
• P(2H) = P(1T) = Number of outcomes with two heads /Total Outcomes = 3/8
• P(3H) = P(0T) = Number of outcomes with three heads/Total Outcomes = 1/8

Dice Roll Probability

Many games use dice to decide the moves of players across the games. A dice has six possible outcomes and the outcomes of a dice is a game of chance and can be obtained by using the concepts of probability. Some games also use two dice, and there are numerous probabilities that can be calculated for outcomes using two dice. Let us now check the outcomes, their probabilities for one dice and two dice respectively.

Rolling One Dice

The total number of outcomes on rolling a die is 6, and the sample space is {1, 2, 3, 4, 5, 6}. Here we shall compute the following few probabilities to help in better understanding the concept of probability on rolling one dice.

• P( Even Number ) = Number of even number outcomes/Total Outcomes = 3/6 = 1/2
• P( Odd Number ) = Number of odd number outcomes/Total Outcomes = 3/6 = 1/2
• P( Prime Number ) = Number of prime number outcomes/Total Outcomes = 3/6 = 1/2

Rolling Two Dice

The total number of outcomes on rolling two dice is 6 2 = 36. The following image shows the sample space of 36 outcomes on rolling two dice.

Let us check a few probabilities of the outcomes from two dice. The probabilities are as follows.

• Probability of getting a doublet(Same number) = 6/36 = 1/6
• Probability of getting a number 3 on at least one dice = 11/36
• Probability of getting a sum of 7 = 6/36 = 1/6

As we see, when we roll a single die, there are 6 possibilities. When we roll two dice, there are 36 (= 6 2 ) possibilities. When we roll 3 dice we get 216 (= 6 3 ) possibilities. So a general formula to represent the number of outcomes on rolling 'n' dice is 6 n .

Probability of Drawing Cards

A deck containing 52 cards is grouped into four suits of clubs, diamonds, hearts, and spades. Each of the clubs, diamonds, hearts, and spades have 13 cards each, which sum up to 52. Now let us discuss the probability of drawing cards from a pack. The symbols on the cards are shown below. Spades and clubs are black cards. Hearts and diamonds are red cards.

The 13 cards in each suit are ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king. In these, the jack, the queen, and the king are called face cards. We can understand the card probability from the following examples.

• The probability of drawing a black card is P(Black card) = 26/52 = 1/2
• The probability of drawing a hearts card is P(Hearts) = 13/52 = 1/4
• The probability of drawing a face card is P(Face card) = 12/52 = 3/13
• The probability of drawing a card numbered 4 is P(4) = 4/52 = 1/13
• The probability of drawing a red card numbered 4 is P(4 Red) = 2/52 = 1/26

Probability Theorems

The following theorems of probability are helpful to understand the applications of probability and also perform the numerous calculations involving probability.

Theorem 1: The sum of the probability of happening of an event and not happening of an event is equal to 1. P(A) + P(A') = 1.

Theorem 2: The probability of an impossible event or the probability of an event not happening is always equal to 0. P(ϕ) = 0.

Theorem 3: The probability of a sure event is always equal to 1. P(A) = 1

Theorem 4: The probability of happening of any event always lies between 0 and 1. 0 < P(A) < 1

Theorem 5: If there are two events A and B, we can apply the formula of the union of two sets and we can derive the formula for the probability of happening of event A or event B as follows.

P(A∪B) = P(A) + P(B) - P(A∩B)

Also for two mutually exclusive events A and B, we have P( A U B) = P(A) + P(B)

Bayes' Theorem on Conditional Probability

Bayes' theorem describes the probability of an event based on the condition of occurrence of other events. It is also called conditional probability . It helps in calculating the probability of happening of one event based on the condition of happening of another event.

For example, let us assume that there are three bags with each bag containing some blue, green, and yellow balls. What is the probability of picking a yellow ball from the third bag? Since there are blue and green colored balls also, we can arrive at the probability based on these conditions also. Such a probability is called conditional probability.

The formula for Bayes' theorem is \(\begin{align}P(A|B) = \dfrac{ P(B|A)·P(A)} {P(B)}\end{align}\)

where, \(\begin{align}P(A|B) \end{align}\) denotes how often event A happens on a condition that B happens.

where, \(\begin{align}P(B|A) \end{align}\) denotes how often event B happens on a condition that A happens.

\(\begin{align}P(A) \end{align}\) the likelihood of occurrence of event A.

\(\begin{align}P(B) \end{align}\) the likelihood of occurrence of event B.

Law of Total Probability

If there are n number of events in an experiment, then the sum of the probabilities of those n events is always equal to 1.

P(A 1 ) + P(A 2 ) + P(A 3 ) + … + P(A n ) = 1

Important Notes on Probability:

• Probability is a measure of how likely an event is to happen.
• Probability is represented as a fraction and always lies between 0 and 1.
• An event can be defined as a subset of sample space.
• The sample of throwing a coin is {head, tail} and the sample space of throwing dice is {1, 2, 3, 4, 5, 6}.
• A random experiment cannot predict the exact outcomes but only some probable outcomes.

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Probability Examples

Example 1: What is the probability of getting a sum of 10 when two dice are thrown?

There are 36 possibilities when we throw two dice.

The desired outcome is 10. To get 10, we can have three favorable outcomes.

{(4,6),(6,4),(5,5)}

Probability of an event = number of favorable outcomes/ sample space

Probability of getting number 10 = 3/36 =1/12

Answer: Therefore the probability of getting a sum of 10 is 1/12.

Example 2: In a bag, there are 6 blue balls and 8 yellow balls. One ball is selected randomly from the bag. Find the probability of getting a blue ball.

Let us assume the probability of drawing a blue ball to be P(B)

Number of favorable outcomes to get a blue ball = 6

Total number of balls in the bag = 14

P(B) = Number of favorable outcomes/Total number of outcomes = 6/14 = 3/7

Answer: Therefore the probability of drawing a blue ball is 3/7.

Example 3: There are 5 cards numbered: 2, 3, 4, 5, 6. Find the probability of picking a prime number, and putting it back, you pick a composite number.

The two events are independent. Thus we use the product of the probability of the events.

P(getting a prime) = n(favorable events)/ n(sample space) = {2, 3, 5}/{2, 3, 4, 5, 6} = 3/5

p(getting a composite) = n(favorable events)/ n(sample space) = {4, 6}/{2, 3, 4, 5, 6}= 2/5

Thus the total probability of the two independent events = P(prime) × P(composite)

= 3/5 × (2/5)

Answer: Therefore the probability of picking a prime number and a prime number again is 6/25.

Example 4: Find the probability of getting a face card from a standard deck of cards using the probability equation.

Solution: To find: Probability of getting a face card Given: Total number of cards = 52 Number of face cards = Favorable outcomes = 12 Using the probability formula, Probability = (Favorable Outcomes)÷(Total Favourable Outcomes) P(face card) = 12/52 m = 3/13

Answer: The probability of getting a face card is 3/13

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Practice Questions on Probability

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FAQs on Probability

What is the meaning of probability in statistics.

Probability is a branch of math which deals with finding out the likelihood of the occurrence of an event. Probability measures the chance of an event happening and is equal to the number of favorable events divided by the total number of events. The value of probability ranges between 0 and 1, where 0 denotes uncertainty and 1 denotes certainty.

How to Find Probability?

The probability can be found by first knowing the sample space of the outcomes of an experiment. A probability is generally calculated for an event (x) within the sample space. The probability of an event happening is obtained by dividing the number of outcomes of an event by the total number of possible outcomes or sample space.

What are the Three Types of Probability?

The three types of probabilities are theoretical probability, experimental probability, and axiomatic probability. The theoretical probability calculates the probability based on formulas and input values. The experimental probability gives a realistic value and is based on the experimental values for calculation. Quite often the theoretical and experimental probability differ in their results. And the axiomatic probability is based on the axioms which govern the concepts of probability.

How To Calculate Probability?

The probability of any event depends upon the number of favorable outcomes and the total outcomes. Finding probability is finding the ratio of the number of favorable outcomes to the total outcomes in that sample space. It is expressed as, Probability of an event P(E) = (Number of favorable outcomes) ÷ (Number of Elements in Sample space).

What is Conditional Probability?

The conditional probability predicts the happening of one event based on the happening of another event. If there are two events A and B, conditional probability is a chance of occurrence of event B provided the event A has already occurred. The formula for the conditional probability of happening of event B, given that event A, has happened is P(B/A) = P(A ∩ B)/P(A).

What is Experimental Probability?

The experimental probability is based on the results and the values obtained from the probability experiments. Experimental probability is defined as the ratio of the total number of times an event has occurred to the total number of trials conducted. The results of the experimental probability are based on real-life instances and may differ in values from theoretical probability.

What is a Probability Distribution?

The two important probability distributions are binomial distribution and Poisson distribution. The binomial distribution is defined for events with two probability outcomes and for events with a multiple number of times of such events. The Poisson distribution is based on the numerous probability outcomes in a limited space of time, distance, sample space. An example of the binomial distribution is the tossing of a coin with two outcomes, and for conducting such a tossing experiment with n number of coins. A Poisson distribution is for events such as antigen detection in a plasma sample, where the probabilities are numerous.

How are Probability and Statistics Related?

The probability calculates the happening of an experiment and it calculates the happening of a particular event with respect to the entire set of events. For simple events of a few numbers of events, it is easy to calculate the probability. But for calculating probabilities involving numerous events and to manage huge data relating to those events we need the help of statistics . Statistics helps in rightly analyzing

How Probability is Used in Real Life?

Probability has huge applications in games and analysis. Also in real life and industry areas where it is about prediction we make use of probability. The prediction of the price of a stock, or the performance of a team in cricket requires the use of probability concepts. Further, the new technology field of artificial intelligence is extensively based on probability.

Where Do We Use the Probability Formula In Our Real Life?

The following activities in our real-life tend to follow the probability equation:

• Weather forecasting
• Playing cards
• Voting strategy in politics
• Rolling a dice.
• Pulling out the exact matching socks of the same color
• Chances of winning or losing in any sports.

How was Probability Discovered?

The use of the word "probable" started first in the seventeenth century when it was referred to actions or opinions which were held by sensible people. Further, the word probable in the legal content was referred to a proposition that had tangible proof. The field of permutations and combinations, statistical inference, cryptoanalysis, frequency analysis have altogether contributed to this current field of probability.

What is the Conditional Probability Formula?

The conditional probability depends upon the happening of one event based on the happening of another event. The conditional probability formula of happening of event B, given that event A, has already happened is expressed as P(B/A) = P(A ∩ B)/P(A).

Theoretical Probability & Experimental Probability

Related Pages Probability Tree Diagrams Probability Without Replacement Probability Word Problems More Lessons On Probability

In these lessons, we will look into experimental probability and theoretical probability.

The following table highlights the difference between Experimental Probability and Theoretical Probability. Scroll down the page for more examples and solutions.

How To Find The Experimental Probability Of An Event?

Step 1: Conduct an experiment and record the number of times the event occurs and the number of times the activity is performed.

Step 2: Divide the two numbers to obtain the Experimental Probability.

How To Find The Theoretical Probability Of An Event?

The Theoretical Probability of an event is the number of ways the event can occur (favorable outcomes) divided by the number of total outcomes.

What Is The Theoretical Probability Formula?

The formula for theoretical probability of an event is

Experimental Probability

One way to find the probability of an event is to conduct an experiment.

Example: A bag contains 10 red marbles, 8 blue marbles and 2 yellow marbles. Find the experimental probability of getting a blue marble.

Solution: Take a marble from the bag. Record the color and return the marble. Repeat a few times (maybe 10 times). Count the number of times a blue marble was picked (Suppose it is 6).

How to find and use experimental probability?

The following video gives another example of experimental probability.

How the results of the experimental probability may approach the theoretical probability?

Example: The spinner below shows 10 equally sized slices. Heather spun 50 times and got the following results. a) From Heather’s’ results, compute the experimental probability of landing on yellow. b) Assuming that the spinner is fair, compute the theoretical probability of landing in yellow.

Theoretical Probability

We can also find the theoretical probability of an event.

Example: A bag contains 10 red marbles, 8 blue marbles and 2 yellow marbles. Find the theoretical probability of getting a blue marble.

Solution: There are 8 blue marbles. Therefore, the number of favorable outcomes = 8. There are a total of 20 marbles. Therefore, the number of total outcomes = 20

Example: Find the probability of rolling an even number when you roll a die containing the numbers 1-6. Express the probability as a fraction, decimal, ratio and percent.

Solution: The possible even numbers are 2, 4, 6. Number of favorable outcomes = 3. Total number of outcomes = 6

Comparing Theoretical And Experimental Probability

The following video gives an example of theoretical and experimental probability.

Example: According to theoretical probability, how many times can we expect to land on each color in a spinner, if we take 16 spins? Conduct the experiment to get the experimental probability.

We will then compare the Theoretical Probability and the Experimental Probability.

The following video shows another example of how to find the theoretical probability of an event.

A spinner is divided into eight equal sectors, numbered 1 through 8. a) What is the probability of spinning an odd numbers? b) What is the probability of spinning a number divisible by 4? b) What is the probability of spinning a number less than 3?

A spinner is divided into eight equal sectors, numbered 1 through 8. a) What is the probability of spinning a 2? b) What is the probability of spinning a number from 1 to 4? b) What is the probability of spinning a number divisible by 2?

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• Math Article
• Theoretical Probability

Every one of us would have encountered multiple situations in life where we had to take a chance or risk. Depending on the situation, it can be predicted up to a certain extent if a particular event is going to take place or not. This chance of occurrence of a particular event is what we study in probability. In our everyday life, we are more accustomed to the word ‘chance’ as compared to the word ‘probability’. Since Mathematics is all about quantifying things, the theory of probability basically quantifies these chances of occurrence or non-occurrence of certain events. In this article, we are going to discuss what is probability and its two different types of approaches with examples.

What is Probability?

In Mathematics, the probability is a branch that deals with the likelihood of the occurrences of the given event. The probability value is expressed between the range of numbers from 0 to 1. The three basic rules connected with the probability are addition, multiplication, and complement rules.

Theoretical Probability Vs Experimental Probability

Probability theory can be studied using two different approaches:

• Experimental Probability

Theoretical Probability Definition

Theoretical probability is the theory behind probability. To find the probability of an event using theoretical probability, it is not required to conduct an experiment. Instead of that, we should know about the situation to find the probability of an event occurring. The theoretical probability is defined as the ratio of the number of favourable outcomes to the number of possible outcomes.

Probability of Event P(E) = No. of. Favourable outcomes/ No. of. Possible outcomes.

Experimental Probability Definition

The experimental probability also is known as an empirical probability, is an approach that relies upon actual experiments and adequate recordings of occurrence of certain events while the theoretical probability attempts to predict what will happen based upon the total number of outcomes possible. The experimental Probability is defined as the ratio of the number of times that event occurs to the total number of trials.

Probability of Event P(E) = No. of. times that event occurs/ Total number of trials

The basic difference between these two approaches is that in the experimental approach; the probability of an event is based on what has actually happened by conducting a series of actual experiments, while in theoretical approach; we attempt to predict what will occur without actually performing the experiments.

Theoretical probability Example

Find the probability of rolling a 5 on a fair die

To find the probability of getting 5 while rolling a die, an experiment is not needed. We know that there are 6 possible outcomes when rolling a die. They are 1, 2, 3, 4, 5, 6.

Therefore, the probability  is,

P(E) = 1/6.

Hence, the probability of getting 5 while rolling a fair die is 1/6.

Comparing Experimental and Theoretical probability

Suppose in a cricket match tournament you are the captain of your team. Now, you are on the pitch and umpire tosses a fair coin. Can you predict the consequence or the outcome when the coin is still in the air? No, that is not possible. In this particular situation, tossing a coin in terms of probability is known as an experiment; this experiment is a random experiment since the result is unknown. Therefore, experiments which do not have a fixed result are known as random experiments. The consequence of such experiments is unknown. The result obtained after a random experiment has occurred is known as the outcome of that experiment. In this case, the possible outcomes are Head or Tail. Each outcome of an experiment or a collection of outcomes constitutes an event. If each outcome of an experiment has an equal chance of occurrence then these outcomes are equally likely. As in the example of tossing a fair coin, the chances of occurrence of heads and tails are equally likely. The entire possible set of outcomes of any experiment represents the sample space related to that experiment. The sample space related to any event is represented as S.

To determine the likelihood of random experiments they are repeated several times. An experiment is repeated a fixed number of times and each repetition is known as a trial.

The ratio of a number of favourable outcomes to the number of total outcomes is defined as the probability of occurrence of any event P(E) when the outcomes are equally likely.

For any event the probability of its occurrence always lies between 0 and 1, i.e. 0 < P(E) <1. Also, if an event is sure to happen then its probability is 1 and if it is impossible to occur then its probability is 0. If P(E) is the probability of occurrence of any event and P(E)’ is the probability of non-occurrence of that event then;

P(E) + P(E)’ = 1

It has been observed that the experimental probability of an event approaches to its theoretical probability if the number of trials of an experiment is very large. But the question arises; what is the necessity to study theoretical probability when real-life experiments can be conducted? The answer is pretty simple since in a lot of cases, actually conducting many experiments is either not achievable or it’s too expensive.

Consider the experiment of tossing a coin or drawing a card from a deck of cards, this can be repeated a huge number of times to get a better result. But in situations, like to find the probability of failure of a satellite launch, experiments cannot be conducted multiple numbers of times because launching a satellite using rockets multiple numbers of times is neither feasible nor practical. In such cases, it becomes very crucial to make certain assumptions and based on those assumptions theoretical probability is calculated, which is very useful in such cases, especially in a lot of applications where we cannot perform the experiment.

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Mastering Theoretical and Experimental Probability Comparisons Dive into the world of probability with our comprehensive guide. Learn to calculate, compare, and apply theoretical and experimental probability in real-world scenarios. Enhance your statistical skills today!

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• Experimental probability VS. Theoretical probability

What is the experimental probability of both coins landing on heads?

• Calculate the theoretical probability of both coins landing on heads.
• Compare the theoretical probability and experimental probability.
• What can Jessie do to decrease the difference between the theoretical probability and experimental probability?

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What is experimental probability?

In math, when we deal with probability , we may be asked for the experimental probability of an experiment. What this means is that they're looking for the probability of something happening based off the results of an actual experiment. This is the experimental probability definition.

So for example, if you're asked for the probability of getting heads after flipping a coin 10 times, the experimental probability will be the number of times you got heads after flipping a coin 10 times. Let's say that you got 6 heads out of your 10 throws. Then your experimental probability is 6/10, or 60%.

For theoretical probability, it doesn't require you to actually do the experiment and then look at the results. Instead, the theoretical probability is what you expect to happen in an experiment (the expected probability). This is the theoretical probability definition.

In the case of the coin flips, since there's 2 sides to a coin and there's an equal chance that either side will land when you flip it, the theoretical probability should be 1 2 \frac{1}{2} 2 1 ​ or 50%.

Why is there a difference in theoretical and experimental probability? The relationship between the two is that you'll find if you do the experiment enough times, the experimental probability will get closer and closer to the theoretical probability's answer. You can try this out yourself with a coin. You likely won't get exactly 50% for both heads and tails from your first 10 throws, but as you throw a coin 50 times or even 100 times, you'll see the experimental probability's answer getting closer to 50%.

We'll now see how experimental and theoretical probability works with these questions.

Question 1a: Two coins are flipped 20 times to determine the experimental probability of landing on heads versus tails. The results are in the chart below:

We are looking for the experimental probability of both coins landing on heads. Looking at the table in the question, we know that there were 4 out of 20 trials in which both coins landed on heads. So the experimental probability is 4 20 \frac{4}{20} 20 4 ​ , which equals to 1 5 \frac{1}{5} 5 1 ​ (20%) after simplifying the fraction

Question 1b: Calculate the theoretical probability of both coins landing on heads.

Now, we are looking for the theoretical probability. First, there are 4 possible outcomes (H,H), (H, T), (T,H), (T, T). 1 out of the 4 possible outcomes has both coins land on heads. So, the theoretical probability is 1 4 \frac{1}{4} 4 1 ​ or 25%

Question 1c: Compare the theoretical probability and experimental probability.

From the previous parts, we know that the experimental probability of both coins landing on head equals 20%, while in theory, there should be a 25% chance that both coins lands on head. Therefore, the theoretical probability is higher than the experimental probability.

Question 1d:

What can we do to reduce the difference between the experimental probability and theoretical probability? We can simply continue the experimental by flipping the coin for many more times —say, 20,000 times. When more trials are performed, the difference between experimental probability and theoretical probability will diminish. The experimental probability will gradually get closer to the value of the theoretical probability. In this case, the experimental probability will get closer to 25% as the coins is tossed over more times.

If you're looking for more experimental vs.theoretical probability examples, feel free to try out this question . It'll require you to do some hands-on experimentation!

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Theoretical Probability Definition and Examples

Probability > Theoretical Probability

What is Theoretical Probability?

The study of probability can be divided into two areas:

• Theoretical Probability is the theory behind probability.
• Experimental (empirical) probability is probability calculated during experiments , direct observation, experience, or practice.

With theoretical probability, you don’t actually conduct an experiment (i.e. roll a die or conduct a survey). Instead, you use your knowledge about a situation, some logical reasoning, and/or known formula to calculate the probability of an event happening. It can be written as the ratio of the number of favorable events divided by the number of possible events. For example, if you have two raffle tickets and 100 tickets were sold:

• Number of favorable outcomes: 2
• Number of possible outcomes: 100
• Ratio = number of favorable outcomes / number of possible outcomes = 2/100 = .5.

A theoretical probability distribution is a known distribution like the normal distribution , gamma distribution , or one of dozens of other theoretical distributions.

Theoretical Probability Example

Step 2: Figure out the probability. The entire sample space is made up of 36 possible rolls. There are 9 rolls that result in a 7, so the answer is: 9/36 = .25.

Why not just Conduct Experiments all the Time?

There are several reasons why the field of Theoretical Probability exists. Sometimes, conducting an experiment isn’t possible for practical or financial reasons. For example, you might be studying a rare genetic trait in salamanders and you want to know what the probability of any one salamander having the rare trait is. If you don’t have access to all of the salamanders on the planet, you won’t be able to conduct an experiment so you’ll have to rely on theory to give you the answer. Theoretical probability is also used in many areas of science where direct experimentation isn’t possible. For example, probabilities involving subatomic particles or abstract structures like vector spaces .

LeBlanc, D. (2004). Statistics: Concepts and Applications for Science, Illustrated Edition. Jones & Bartlett.

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Probability in Maths

Probability is a field of mathematics that deals with events and quantifies their likelihood of occurring with numerical values ranging from 0 to 1. Higher probabilities indicate a greater chance of the event happening. It is expressed as a number between 0 and 1.

In this article, we will discuss what is probability in maths, its definition, formula, types, theorems, and solved examples on probability.

Table of Content

What is Probability in Maths?

Probability definition, probability theory, terms in probability, probability of an event, types of events in probability, probability formula, probability tree diagram, types of probability, theoretical probability, experimental probability, axiomatic probability, probability theorems, bayes’ theorem on conditional probability , bayes theorem formula, probability density function, common probability problems, probability examples and solutions.

Probability is the measure of the likelihood that an event will occur. It is quantified as a number between 0 and 1. The values of probability range from 0 to 1, with 0 telling us improbability and 1 denoting certainty.

For calculating probability, we simply divide the number of favorable outcomes by the total number of outcomes.

Probability is a concept within mathematics that quantifies the likelihood of an event occurring. It assigns a numerical value between 0 and 1 to an event, where 0 indicates impossibility and 1 indicates certainty.

Probability theory is a branch of mathematics that deals with the interpretation of random events and the likelihood of these events occurring. Probability theory starts with basic concepts such as random experiments, sample spaces, events, and the probability of events.

A random experiment is any process or action that results in one of several possible outcomes, like tossing a coin or rolling a die. The sample space is the set of all possible outcomes of an experiment, and an event is a specific outcome or a set of outcomes.

Here are some basic concepts or terminologies used in probability:

Probability of an event is a measure of the likelihood that the event will occur, expressed as a number between 0 and 1. An event with a probability of 1 is considered certain to happen, while an event with a probability of 0 is certain not to happen.

Various types of events in probability are:

• Equally Likely Events
• Exhaustive Events
• Mutually Exclusive Events
• Complimentary Events

Let’s discuss different types of events in probability.

Equally Likely Events : Equally likely events are those whose chances or probabilities of happening are equal. Both events are not related to one another. For example, t here are equal possibilities of receiving either a head or a tail when we flip a coin.

Exhaustive Events : We call an event exhaustive when the set of all experiment results is the same as the sample space.

Mutually Exclusive Events : Mutually exclusive Events cannot occur at the same time. For instance, the weather may be hot or chilly simultaneously. We can’t have the same weather at the same time.

Sample Space Illustration in Probability

Complimentary Events : Non-occurring events. The complement of an event A is the event, not A(or A’). 

P(E) = Number of Favorable Outcomes / Total Number of Outcomes 

Note: where P(E) denotes the probability of an event E. 

This formula defines the possibility of an event occurring. It is defined as the ratio of the number of favourable outcomes to the total number of outcomes.  

A tree diagram in probability is a graphic representation that helps us in determining the likely outcomes that is whether an event will occur or not. It helps us understand the all possibilities of an event and which possibilities can occur and cannot occur.

These are 3 major types of probability:

It is focused on the likelihood of anything occurring. The reasoning behind probability is the foundation of scientific probability .

If a coin is flipped, the statistical chance of having a head is 1/2. An event’s statistical chance is the chance that the occurrence will occur. It is determined by dividing the number of desirable outcomes by the total number of outcomes is thus termed theoretical probability.

It is founded on the results of an experiment. The experimental chance can be determined by dividing the total number of trials by the number of potential outcomes.

For example , if a coin is flipped ten times and heads are reported six times, the experimental chance of heads is 6/10 or 3/5.

A collection of laws or axioms that apply to all forms is established in axiomatic probability. Kolmogorov developed these axioms, which are known as Kolmogorov’s three axioms . Axiomatic Approach to Probability quantifies the likelihood of events occurring or not occurring.

Some of the most important theorems of probability are :

Bayes’ Theorem defines the probability of an event based on the condition of occurrence of other events. It is also called conditional probability . It helps in calculating the probability of occurrence of one event based on the condition that another event is also taking place or occurring.

For example , let us assume you have two bags of marbles in various colours. Bag A has three red and two blue marbles, while bag B has one red and four blue marbles. You pick one of the bags at random, then you pick a marble at random from that bag. Given that you selected a red marble, what is the probability that you selected bag A? Such probability is called conditional probability.

Following is the Bayes’ Theorem Formula:

P(A|B) = P(B|A)⋅P(A) P(B)

• P(A|B) denotes how often event A happens on a condition that B happens.
• P(B|A) denotes how often event B happens on a condition that A happens.
• P(A) is the probability of event A occurring.
• P(B) is the probability of event B occurring.

P robability density function is defined as a function that contains all the possible outcomes of any given situation.

It is used to specify the probability of a continuous random variable falling within a particular range of values. Unlike discrete random variables which have probabilities associated with specific points, continuous random variables require a function to define their probabilities over an interval. It is also known as the Probability Distribution Function .  

We will now see some common probability problems that are given in school tests. This will prepare you for the questions and you can understand the methodology to solve them.

1. Probability of Tossing Coin

Now let us take into account the case of coin tossing to understand probability in a better way. 

A) Tossing a Coin

A single coin when flipped has two possible outcomes, a head or a tail. The definition of probability when applied here to find the probability of getting a head or getting a tail. 

The total number of possible outcomes = 2

Sample Space = {H, T} H: Head, T: Tail P(H) = Number of Heads/ Total Number of outcomes = 1/2 P(T) = Number of Tails/ Total Number of outcomes = 1/2

B) Tossing Two Coins

There are a total of four possible results when tossing two coins. We can calculate the probability of two heads or two tails using the formula.

The probability of getting two tails can be calculated as :

Total number of outcomes = 4 

Sample Spac e = {(H,H),(H,T),(T,H),(T,T)} P(2T) = P(0H) = Number of outcomes with two tails/ Total number of outcomes = 1/4 P(1H) = P(1T) = Number of outcomes with one head/ Total number of outcomes = 2/4 = 1/2

C ) Probability of Tossing Three Coins

The number of total outcomes on tossing three coins is 2 3 = 8. For these outcomes, we can find the probability of various cases such as getting one tail, two tails, three tails, and no tails, and similarly can be calculated for several heads. 

Total number of outcomes = 2 3 = 8

Sample space = {(H, H, H), (H, H, T),(H, T, H), (T, H, H), (T, T, H), (T, H, T), (H, T, T), (T, T, T)}. P(3T) = P(0 H) = Number of outcomes with three tails/ Total Number of outcomes = 1/8

2. Probability of Rolling Dice

Various games use dice to decide the movements of the player during the games. A dice has six outcomes. Some games are played using two dice. Now let us calculate the outcomes, and their probabilities for one dice and two dice respectively. 

Rolling One Dice

The number of outcomes is 6 when a die is rolled and the sample space is = {1, 2, 3, 4, 5, 6}.

Let us now discuss some cases.

P(Even Number) = Number of outcomes in which even number occur/Total Outcomes = 3/6 = 1/2 P(Prime Number) = Number of prime number outcomes/ Total Outcomes = 3/6 = 1/2

Rolling Two Dice

The number of total outcomes, when two dice are rolled, is 6 2 = 36.

Probability of Rolling Two Dice

Let us check a few cases in the above example,

Probability of getting a doublet(Same number) = 6/36 = 1/6 Probability of getting a number 3 on at least one dice = 11/36

3. Probability of Cards

Spades, clubs, diamonds, and hearts make up the four suits that form a deck of 52 playing cards. There are a total of 52 cards, with 13 in each of the four suits (clubs, diamonds, hearts, and spades). The symbols for the cards are listed below.

Black cards are spades and clubs. Red cards are hearts and diamonds. Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, queen, and King are the 13 cards that make up each suit. The jack, queen, and king are referred to as face cards in these.

Probability of getting a black card is P(Black Card) = 26/52 = 1/2 Probability of getting a heart is P(Heart) = 13/52 = 1/4 Probability of getting a face card is P(Face card) = 12/52 = 3/13

Articles related to Probability:

• Conditional Probability and Independence
• Multiplication Theorem
• Dependent and Independent Event
• Binomial Random Variables and Binomial Distribution
• Binomial Mean and Standard Deviation
• Bernoulli Trials and Binomial Distribution
• Discrete Random Variables
• Expected Value

We have provided you with some probability problems with their solutions.

Example 1: There are 8 balls in a container, 4 are red, 1 is yellow and 3 are blue. What is the probability of picking a yellow ball?

Probability of picking a ball

Probability is equal to the number of yellow balls in the container divided by the total number of balls in the container, i.e. 1/8. 

Example 2: A dice is rolled. What is the probability that an even number has been obtained?

Probability of Rolling a Dice

When fair six-sided dice are rolled, there are six possible outcomes: 1, 2, 3, 4, 5, or 6. Out of these, half are even (2, 4, 6) and half are odd (1, 3, 5). Therefore, the probability of getting an even number is: P(even) = number of even outcomes / total number of outcomes  P(even) = 3 / 6 P(even) = 1/2

Example 3: A bag contains 4 white, 5 red, and 6 blue balls. Three balls are drawn at random from the bag. The probability that all of them are red, is:

Let S be the sample space. Then, n(S) = Number of ways of drawing 3 balls out of 15 n(S) = 15 C 3 = 455 Let E = event of getting all the 3 red balls.  n(E) = 5 C 3 = 10 P(E) = n(E)/n(S) = 10/455 = 2/91.

Example 4: In a class, there are 10 girls and 15 boys, what is the probability that 1 girl and 2 boys are selected?

Let S be the sample space. Then, n(S) = Number of ways of selecting 3 children out of 25 n(S) = 25 C 3 n(S) = 2300.  Let  E= event of selecting 1 girl and 2 boys.  n(E) = 10 C 1 * 15 C 2 = 1050 P(E) = n(E)/n(S) = 1050/2300 = 21/46. 

Conclusion – Probability in Maths

Probability in Maths is the likelihood that an event will occur. It’s similar to picking whether your favorite team will win a game or whether it will rain tomorrow. Probability is the study of the likelihood of an event or the possible outcomes under specific circumstances. For instance, we might be interested in finding out how often it is to draw a red ball from a bag of colored balls or what the odds are of flipping a coin and receiving heads. We may make sense of uncertainty and well-informed decisions by using probability to compare the chances of various scenarios. It’s a helpful tool in many fields, such as evaluating risks in banking, forecasting game outcomes, and even interpreting data in science.

FAQs on Probability in Maths

Probability is a branch of mathematics that is a measure of how probable an incident is to occur. For example, when a coin is tossed in the air, there are two possible outcomes head or tail. 

How many Types of Probability are there?

There are three types of probability: Theoretical Probability Experimental Probability Axiomatic Probability

What is the Probability Formula?

P(E) = Number of Favorable Outcomes /Total Number of Outcomes

What is Conditional Probability?

Conditional probability is a measure of the probability of an event occurring, given that another event has already occurred. This concept is fundamental in probability theory and statistics, and it’s denoted as P(A|B), which is read as “the probability of A given B.”

How do I Calculate Probability?

To Calculate the probability of an event, find the ratio of the number of favorable outcomes to the total outcomes in that sample space. It is expressed as, Probability of an event P(E) = (Number of favorable outcomes) ÷ (Number of Elements in Sample space).

What is Probability of Impossible Event?

The probability of an impossible event is zero.

What is Probability Density Function?

A Probability Density Function (PDF) represents the likelihood of a continuous random variable falling within a particular range of values. It is a statistical concept used in probability theory and statistics.

What is Probability Mass Function ?

A Probability Mass Function (PMF) represents the probability distribution of a discrete random variable, which can take on a finite or countably infinite number of possible values.

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Axiomatic Approach to Probability: Application, Example

Axiomatic Approach to Probability: Probability is the certainty that an event will occur. It is a field of Mathematics dealing with numerical explanations of the chance of an event occurring or the truth of a statement. Probability can only be applied to experiments in which the complete number of outcomes is known, i.e., the idea of probability cannot be applied unless the total number of outcomes of an experiment is known.

Hence, we need to know the total number of possible outcomes of the experiment to apply probability in everyday situations. Axiomatic probability is an approach to expressing the probability of an event occurring. Several axioms or rules are predefined before assigning probabilities. This is done to quantify the event and make calculating the occurrence or non-occurrence of events easier.

Axiomatic Approach to Probability: Overview

One of the techniques to describe the probability of an event occurring is to use an axiomatic approach. We are familiar with random experiments, sample space, and occurrences related to probability experiments. We use various terms regarding the probability of events occurring in our daily lives, such as weather forecasts, buying and selling. Probability theory measures the probability of events occurring or not occurring.

The theoretical probability of an event is the ratio of the number of outcomes to the total number of outcomes. In contrast, the axiomatic approach is quite different. In this approach, some axioms or rules are depicted to assign probabilities.

Axiomatic probability is a theory that unifies probability. It establishes a set of axioms (laws) that apply to all types of probability, including frequentist and classical probabilities. Based on Kolmogorov’s three axioms, these laws establish the starting points of mathematical probability.

Classical and Axiomatic Approach

This method is based on the assumption that all outcomes are equally likely. Suppose an event can occur in \(n\) different ways out of a total of \(N\) possibilities. The earliest probability is the classical probability, typically applied to simple scenarios such as a gambling game. Assume a random experiment (such as a roll of the dice) yields a finite number, \(n\) of equally likely possibilities. If  \(m\) of the outcomes have a unique feature, the probability of that characteristic is the ratio \(\frac{m}{n}\). This is effective for examining dice throws and card picks, but it is less useful in complex scenarios.

Frequentist Probability

This method of estimating probability is more general. It does not assume that all of the possible outcomes are equally likely. We repeat the experiment numerous, let’s say \(M\) times, when the results are not equally likely. Then, count how many times that same output occurred, say \(m\). Lastly, come up with an empirical probability estimate. Hence, we use of the relationship shown below. \(P\left( {{\text{event}}} \right) = {\log _{M \to \infty }}\frac{m}{M}\)

The value of empirical probability should only be used when the experiment is performed infinite times.

Finally, both approaches fail to withstand the rigours of Mathematics because the former incorporates the uncertain phrase, “equally likely,” for which there is no mathematical justification. The latter lacks a way to prove that \({\log _{M \to \infty }}\frac{m}{M}\)  will cover some value because no experiment can be repeated infinite times.

The axiomatic approach to probability considers probability as a function associated with any event, even if the event is unlikely.

Statement: The probability of every given event \(X\) must be greater than or equal to zero. Thus, \(0 \leqslant P\left( X \right)\) The probability of an event defined on the sample space is larger than or equal to zero, according to Axiom \(1\). If the sample space has \(n\) points, the impossible event, that is, an event with no sample points, is the empty event on \(S\) with a probability of zero. This event is denoted by \(\emptyset \). \(P\left( \emptyset  \right) = \frac{0}{n}\) \(\therefore \,P\left( \emptyset  \right) = \,0\)

Statement: The set of all the outcomes is known to be the sample space \(S\) of the experiment. This suggests that the chance of every given outcome occurring is \(100\% \) or \(P\left( S \right) = 1\). Intuitively, this suggests a \(100\% \) chance of achieving a specific result whenever this experiment is performed. \(P\left( S \right) = 1\) The probability of \(S\), the sample space, is one, according to Axiom \(2\). That is to say, \(P\left( S \right) = \frac{n}{n}\) \(\therefore P\left( S \right) = 1\)

 Note that if we calculate the likelihood of each event on the sample space \(S\), we call it a probability space.

Furthermore, if we add up the probabilities of all possible simple events on \(S\), we get one or \(100\% \). 

Statement: For the experiments with two outcomes,  \(A\) and \(B\), if \(A\) and \(B\) are mutually exclusive, then \(P\left( {A \cup B} \right) = P\left( A \right) + P\left( B \right)\) The probability of the union of more than one event on \(S\) equals the total of their probabilities, according to Axiom \(3\), assuming that the sequence of events is mutually exclusive. You may recall that if \({A_{1\,}} \cap \,\,{A_2} = \emptyset \), two occurrences \({A_{1\,}}\) and \({A_{2\,}}\) in \(S\) are said to be mutually exclusive. Every feasible pair in the sequence must be mutually exclusive to satisfy the condition.

Conditions of Axiomatic Approach

For the events \({A_1},\,{A_2},\,……{A_n}\) defined in the sample space \(S\). The probability of each event, denoted by \(P\left( {{A_i}} \right)\) where \(i = 1,\,2,….n,\) are numbers that satisfy the following three conditions: 1. \(0 \leqslant P\left( {{A_i}} \right)\) 2. \(P\left( S \right) = 1\,{\text{or }}P\left( {{A_1}} \right) + P\left( {{A_2}} \right) \pm …… + P\left( {{A_n}} \right) = 1\) 3. \(P\left( {{A_1}\, \cup \,{A_2}\, \cup ……{A_n}} \right) = P\left( {{A_1}} \right) + P\left( {{A_2}} \right) + ….P\left( {{A_n}} \right)\) where \({A_1},\,{A_2},\,……{A_n}\) are mutually exclusive events.

Probability of Equally Likely Outcomes

Equally likely events are those that have the same theoretical probability (or likelihood) of occurrence.

Example:  When a die is tossed, each number getting on a die is equally likely to occur. The probability of getting each number on a die is \(\frac{1}{2}\).

Probability of OR Events

The probability of ‘\(A\) or \(B\)’ is calculated as \(P\left( {A\,{\text{or}}\,B} \right) = P\left( A \right) + P\left( B \right)\), where \(A\) and \(B\) are mutually exclusive events.

Example: The events of getting even numbers or odd numbers are mutually exclusive. The probability of such events is \(P\left( {{\text{Even}}\,{\text{or}}\,{\text{Odd}}} \right) = \frac{1}{2} + \frac{1}{2} = 1\).

Probability of Complementary Events

Two events are complementary events when one event occurs if and only if the other does not. The sum of probabilities of two complementary events is one.

Example: Getting a head or a tail is a complementary event.

\(P\left( {{\text{head}}} \right) = 1 – P\left( {{\text{tail}}} \right) = \frac{1}{2}\)

Applications of Axiomatic Approach

• Probability theory is used in risk assessment and modelling in everyday life. Actuarial science is used by the insurance sector and markets to calculate prices and make choices.
• Probability can also be used to examine trends in biology (for example, disease spread) and ecology.
• Probability and ancient game references, market surveys, and player inputs are used to build games.
• In natural language processing, the cache language model and other statistical language models are instances of probability theory applications.

The classical probability of an event is finding the possibility of equally likely outcomes, and it is the ratio of favourable outcomes to the total outcomes.

For checking the non-equally likely outcomes, Andrey Kolmogorov developed Kolmogorov laws in 1933. The axiomatic approach is finding the probability by applying some rules or axioms.

The three laws of Kolmogorov are the law of impossible, sum of elementary events and mutually exclusive events. They tell the probability of an impossible event and sure event, the sum of probabilities of event and also that of mutually exclusive events.

The probability of equally lightly outcomes is same for each outcome. probability of mutually exclusive events is the sum of probabilities of the events.

Solved Examples

 Q1. If an unbiased coin was tossed three times, what is the probability of getting three heads or three tails? Solution: The number of events in a sample space of tossing a coin three times is given by the formula \({2^n}\), where \(n\)is the number of coins tossed. So, total outcomes in a sample space, \({2^3} = 2 \times 2 \times 2 = 8\) \(S = \left\{ {HHH,\,HHT,HTH,HTT,THH,THT,TTH,TTT} \right\}\) The events getting three heads \(\left( A \right) = \left\{ {HHH} \right\}\) The events getting three tails \(\left( B \right) = \left\{ {TTT} \right\}\) Here, the event of getting three heads and three tails are mutually exclusive events. So, probability \( = \frac{{{\text{favorable}}\,{\text{outcomes}}}}{{{\text{total}}\,{\text{number}}\,{\text{of}}\,{\text{outcomes}}}}\) Probability of getting all three as heads or tails is given by \(P\left( {A\,{\text{or}}\,B} \right) = P\left( A \right) + P\left( B \right) = \frac{1}{8} + \frac{1}{8} = \frac{2}{8} = \frac{1}{4}\)

Q2. If \(\frac{2}{{11}}\)  is the probability of an event, what is the probability of the event “not \(A\) ”? Solution: Probability \( = \frac{2}{{11}}\) Sum of probabilities of two complementary events is one. Probability of \(A\) is the complementary event \((A’)\) of the given event. \(P\left( A \right) + P\left( {A’} \right) = 1\) \(P\left( {A’} \right) = 1 – \frac{2}{{11}}\) \( = \frac{{\left( {11 – 2} \right)}}{{11}}\) \(\therefore \,P\left( {A’} \right) = \frac{9}{{11}}\) Hence, the probability of the event \(A’\) is \(\frac{9}{{11}}\).

Q3. If an experiment has exactly the three possible mutually exclusive outcomes \(A, B\) and \(C\) check each case whether the assignment of probability is permissible \(P\left( A \right) = \frac{4}{7},P\left( B \right) = \frac{1}{7},\,P\left( C \right) = \frac{2}{7}\) Solution: Since the experiment has three possible mutually exclusive outcomes \(A, B\) and \(C\), they are exhaustive events. \( \Rightarrow S = A \cup B \cup C\)

Therefore, using the axioms of probability, we get \(P\left( A \right) \geqslant 0,P\left( B \right) \geqslant 0,P\left( C \right) \geqslant 0\,{\text{and}}\,P\left( {A \cup B \cup C} \right) = P\left( A \right) + P\left( B \right)9 + P\left( C \right)\) \( = P\left( S \right) = 1\) \( \Rightarrow \frac{4}{7} + \frac{1}{7} + \frac{2}{7} = \frac{{4 + 1 + 2}}{7} = \frac{7}{7} = 1\) So, the given probabilities are permissible.

Q4. Find out the probability of getting a number less than \(7\)  when a die is tossed.  Solution: We know that possible outcomes when a die is tossed are \(\left\{ {1,\,2,\,3,\,4,\,5,\,6} \right\}\) The numbers less than \(7\) are \(1, 2, 3, 4, 5, 6\) Number of favourable outcomes \(=6\) Total number of outcomes \(=6\) So, the probability of getting a number less than \(7\) is given by \(P\left( {n < 7} \right) = \frac{6}{6}\) \(\therefore \,P\left( {n < 7} \right) = 1\)

Q5. A bag has \(5\) red balls and \(3\) black balls. Find out the probability of getting a ball and verify the second axiom. Solution: Let us define two events, \(R=\) Red ball is picked  \(B=\) Black Ball is picked Probability of getting a red ball is \(P\left( R \right) = \frac{5}{8}\) Probability of getting a black ball is \(P\left( B \right) = \frac{3}{8}\) \(P\left( R \right) + P\left( B \right) = \frac{5}{8} + \frac{3}{8}\) \( = \frac{8}{8}\) \(\therefore P\left( R \right) + P\left( B \right) = \,1\,\) Thus, the second axiom is also satisfied.

Frequently Asked Questions (FAQs)

Q1. What are the 3  axioms of probability? Ans:  For the events \({A_1},\,{A_2},\,……{A_n}\) be events defined on the sample space  S . The three axioms of probability is given by 1. \(0 \leqslant P\left( {{A_i}} \right)\) 2. \(P\left( S \right) = 1\) 3. \(P\left( {{A_1}\, \cup \,{A_2}\, \cup ……{A_n}} \right) = P\left( {{A_1}} \right) + P\left( {{A_2}} \right) + ….P\left( {{A_n}} \right)\) where \({A_1},\,{A_2},\,……{A_n}\) are mutually exclusive events.

Q2. What are two approaches of probability? Ans: The two approaches to probabilities are: 1. Classical approach 2. Axiomatic approach

Q3. What is the meaning of the axiomatic approach? Ans:  The axiomatic approach to probability considers probability as a function associated with any event, even if the event is unlikely, which follows certain rules or axioms.

Q4. How do we calculate the classical probability? Ans:  Classical probability of the \(n\) things of total \(m\) things are given by the ratio of \(m\) to \(n\). \(P\left( {{\text{event}}} \right) = \frac{n}{m}\)

Q5 . Who is well known for the axiomatic approach   in probability? Ans:  Andrey Kolmogorov developed the Kolmogorov Axioms in 1933, which is the bases of the development of axiomatic approach.

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Why does experimental probability approach theoretical probability? Why does it converge only when there are large samples and not when it's small?

I went through Khan Academy's lecture on theoretical and experimental probability . I also read through a Wikipedia article on this but was still not clear. I understand how it approaches (as explained in the video) but unable to understand why experimental probability approaches theoretical probability. What is the reason for this?

I think that the general sense is, if I take a large enough sample, I am going to end up getting the expected mean of the sample. The more experiments I do, the more it converges. Sure, I get that. But why does it converge only when there are large samples and not when it's small?

• probability

• $\begingroup$ Law of Large Numbers. $\endgroup$ –  Tony Hellmuth Commented Jul 5, 2018 at 11:17
• 1 $\begingroup$ @TonyHellmuth: I think the OP is indirectly asking "why" the strong law of large numbers is correct. So an intuitive explanation of the proof of the strong law of large numbers (if that's possible) should suffice $\endgroup$ –  Shirish Commented Jul 5, 2018 at 11:44
• $\begingroup$ I think that the general sense is, if I take a large enough sample, I am going to end up getting the expected mean of the sample. The more experiments I do, the more it converges. Sure, I get that. But why does it converge only when there are large samples and not when it's small? $\endgroup$ –  Sudhanva Narayana Commented Jul 5, 2018 at 11:46
• $\begingroup$ @SudhanvaNarayana: It's better to edit your question accordingly. In its current state, it's not clear enough what you expect from an answer. $\endgroup$ –  Shirish Commented Jul 5, 2018 at 11:48
• 1 $\begingroup$ The fluctuations that almost inevitable happen, are significant in the case of a small number of observations. If the number of observations gets bigger, the fluctuations are less and less significant. This phenomen is best described in the central limit theorem. I suggest that you carefully study this theorem. $\endgroup$ –  Peter Commented Jul 5, 2018 at 11:59

You think you are "unable to understand why experimental probability approaches theoretical probability". We all are. Experimental probability means really throwing physical coins or needles, or picking colored balls from an urn, and counting the various outcomes.

On the other hand probability theory is a mathematical edifice with the purpose to talk coherently about events and processes considered "random". Take throwing a coin as an example. At the beginning we only postulate that in a single throw we see $H$ or $T$ with equal probabilities ${1\over2}$, whatever that means . We then create the idea of independence . This entails that when throwing the coin $n$ times all $2^n$ binary strings over $\{H,T\}$have the same probability ${1\over2^n}$. In this model we don't know which string we shall observe, but we can prove that with high probability we see about ${n\over2}$ $H$s. This means that the model behaves in the way we intuitively think about probabilities. But the model then also has answers to more difficult questions, e.g., how often will we (on average) have to throw the coin in order to see a run of $10$ $H$s.

Concerning "convergence": This notion by its very name requires large numbers of "experiments" within our model. One then, e.g., proves that, for an infinite sequence of coin throws, with probability $1$ the fraction of $H$s converges to ${1\over2}$. Again: Why this seems to be the case also when you throw a coin $10^6$ times in your lab, nobody knows.

But maybe other people think differently. There is a large literature about the "philosophy of probability".

• $\begingroup$ Good answer! +1 $\endgroup$ –  Physiker Commented Mar 29, 2021 at 2:19
• $\begingroup$ Well, if we can prove that for a sufficiently large number of trials, the number of H s becomes $n/2$, doesn't it mean that the truth for experiments meeting theory after infinite experiments is embedded in the fact that we initially considered the coin to have $1/2$ probability of showing heads? $\endgroup$ –  Physiker Commented Mar 29, 2021 at 2:24
• $\begingroup$ @SarthakGirdhar: From the probability ${1\over2}$ for heads, the assumption of independence and the axioms of probability it is easy to prove mathematically that the limit in question is ${1\over2}$. But the "philosophical" problem is: Why does the real physical coin behave like probability theory says? $\endgroup$ –  Christian Blatter Commented Mar 29, 2021 at 8:33
• $\begingroup$ Because there is no reason why it has to favour one side more? I mean, I sense I am understanding it wrong altogether. You are talking in terms of philosophy, and I am talking in terms of what I think is axiomatic, thus subjecting me to prejudice? Would you like to educate me more? $\endgroup$ –  Physiker Commented Mar 29, 2021 at 8:51

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End-to-end probability analysis method for multi-core distributed systems

• Published: 13 September 2024

Cite this article

• Xianchen Shi 1 ,
• Yian Zhu 1 &
• Lian Li 2  

Timing determinism in embedded real-time systems requires meeting timing constraints not only for individual tasks but also for chains of tasks that involve multiple messages. End-to-end analysis is a commonly used approach for solving such problems. However, the temporal properties of tasks often have uncertainty, which makes end-to-end analysis challenging and prone to errors. In this paper, we focus on enhancing the precision and safety of end-to-end timing analysis by introducing a novel probabilistic method. Our approach involves establishing a probabilistic model for end-to-end timing analysis and implementing two algorithms: the maximum data age detection algorithm and the end-to-end timing deadline miss probability detection algorithm. The experimental results indicate that our approach surpasses traditional analytical methods in terms of safety and significantly enhances the capability to detect the probability of missing end-to-end deadlines.

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Acknowledgements

This work was supported by the National Key Research and Development Program of China (2021YFC2802503) and the Key Research and Development Program of Shaanxi Province ( 2019ZDLGY12-07, 2021ZDLGY05-05).

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School of Computer Science, Northwestern Polytechnical University, Xi’an, 710072, Shaanxi, China

Xianchen Shi & Yian Zhu

School of Software, Northwestern Polytechnical University, Xi’an, 710072, Shaanxi, China

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Contributions

Yian Zhu proposed the research idea and guidance, Xianchen Shi proposed the probabilistic analysis method for end-to-end delay and wrote the paper, and Lian Li was responsible for implementing the algorithm and designing the experimental scheme.

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Correspondence to Yian Zhu .

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Shi, X., Zhu, Y. & Li, L. End-to-end probability analysis method for multi-core distributed systems. J Supercomput (2024). https://doi.org/10.1007/s11227-024-06460-8

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Accepted : 14 August 2024

Published : 13 September 2024

DOI : https://doi.org/10.1007/s11227-024-06460-8

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