random experiment what is

Random Experiments

We may perform various activities in our daily existence, sometimes repeating the same actions though we get the same result every time. Suppose, in mathematics, we can directly say that the sum of all interior angles of a given quadrilateral is 360 degrees, even if we don’t know the type of quadrilateral and the measure of each internal angle. Also, we might perform several experimental activities, where the result may or may not be the same even when they are repeated under the same conditions. For example, when we toss a coin, it may turn up a tail or a head, but we are unsure which results will be obtained. These types of experiments are called random experiments.

Random Experiment in Probability

An activity that produces a result or an outcome is called an experiment. It is an element of uncertainty as to which one of these occurs when we perform an activity or experiment. Usually, we may get a different number of outcomes from an experiment. However, when an experiment satisfies the following two conditions, it is called a random experiment.

(i) It has more than one possible outcome.

(ii) It is not possible to predict the outcome in advance.

Let’s have a look at the terms involved in random experiments which we use frequently in probability theory. Also, these terms are used to describe whether an experiment is random or not.

Learn more about sample space here.

What is a Random Experiment?

Based on the definition of random experiment we can identify whether the given experiment is random or not. Go through the examples to understand what is a random experiment and what is not a random experiment.

Is picking a card from a well-shuffled deck of cards a random experiment?

We know that a deck contains 52 cards, and each of these cards has an equal chance to be selected.

(i) The experiment can be repeated since we can shuffle the deck of cards every time before picking a card and there are 52 possible outcomes.

(ii) It is possible to pick any of the 52 cards, and hence the outcome is not predictable before.

Thus, the given activity satisfies the two conditions of being a random experiment.

Hence, this is a random experiment.

Consider the experiment of dividing 36 by 4 using a calculator. Check whether it is a random experiment or not.

(i) This activity can be repeated under identical conditions though it has only one possible result.

(ii) The outcome is always 9, which means we can predict the outcome each time we repeat the operation.

Hence, the given activity is not a random experiment.

Examples of Random Experiments

Below are the examples of random experiments and the corresponding sample space.

Number of possible outcomes = 8

Number of possible outcomes = 36

Number of possible outcomes = 100

Similarly, we can write several examples which can be treated as random experiments.

Playing Cards

Probability theory is the systematic consideration of outcomes of a random experiment. As defined above, some of the experiments include rolling a die, tossing coins, and so on. There is another experiment of playing cards. Here, a deck of cards is considered as the sample space. For example, picking a black card from a well-shuffled deck is also considered an event of the experiment, where shuffling cards is treater as the experiment of probability.

A deck contains 52 cards, 26 are black, and 16 are red.

However, these playing cards are classified into 4 suits, namely Spades, Hearts, Diamonds, and Clubs. Each of these four suits contains 13 cards.

We can also classify the playing cards into 3 categories as:

Aces:  A deck contains 4 Aces, of which 1 of every suit. 

Face cards:  Kings, Queens, and Jacks in all four suits, also known as court cards.

Number cards:  All cards from 2 to 10 in any suit are called the number cards. 

• Spades and Clubs are black cards, whereas Hearts and Diamonds are red.
• 13 cards of each suit = 1 Ace + 3 face cards + 9 number cards
• The probability of drawing any card will always lie between 0 and 1.
• The number of spades, hearts, diamonds, and clubs is the same in every pack of 52 playing cards.

An example problem on picking a card from a deck is given above.

Leave a Comment Cancel reply

Your Mobile number and Email id will not be published. Required fields are marked *

Request OTP on Voice Call

Post My Comment

Register with BYJU'S & Download Free PDFs

Register with byju's & watch live videos.

Random Experiments

A random experiment is a type of experiment that has multiple possible outcomes. Such an experiment can be repeated many times. In probability theory, once the random experiment has been performed multiple times then the experimental probabilities of various outcomes can be calculated.

An example of a random experiment is a Bernoulli trial in which there are exactly two possible outcomes. Any outcome of a random experiment cannot be predicted until the experiment has been performed. In this article, we will learn more about a random experiment, its definition, and various associated examples.

What are Random Experiments in Probability?

A random experiment is a very important part of probability theory. This is because probability theory is based on the assumption that an experiment is random and can be repeated several times under the same condition. An experiment in probability will have a sample space, a set of events as well as the probabilities of occurrence of those events.

Random Experiments Definition

Random experiments can be defined as experiments that can be performed many times under the same conditions and their outcome cannot be predicted with complete certainty. In order words, in a random experiment, all the possible outcomes are known, however, its exact outcome cannot be precisely predicted in advance. There are certain terms associated with random experiments that are given as follows:

• Sample space: A sample space can be defined as the list of all possible outcomes of a random experiment.
• Outcome: An outcome is a possible result of the random experiment.
• Event: An event is a possible outcome of an experiment and forms a subset of the sample space.
• Trial: When a random experiment is repeated many times each one is known as a trial.

Random Experiments Example

Suppose a coin is tossed. The two possible outcomes are getting a head or a tail. The outcome of this experiment cannot be predicted before it has been performed. Furthermore, it can be conducted many times under the same conditions. Thus, tossing a coin is an example of a random experiment.

Another random experiment example is that of rolling a dice . There can be 6 possible outcomes {1, 2, 3, 4, 5, 6}. However, none of the outcomes can be exactly predicted.

How to Find Probability of Random Experiments?

Probability can be defined as the likelihood of occurrence of an outcome of a random experiment. The formula for finding the probability is given as the number of favorable outcomes divided by the total number of possible outcomes of that random experiment. Suppose the probability of getting exactly two heads needs to be determined when a fair coin is tossed twice. The steps to find the probability are as follows:

• Step 1: Determine the sample space of the random experiment or the total number of outcomes. The sample space of a coin tossed twice is given as {HH, HT, TH, TT}. Thus, the total number of outcomes are 4.
• Step 2: Find the number of favorable outcomes. As the probability of getting exactly two heads needs to be determined the number of favorable outcomes is 1.
• Step 3: Apply the probability formula. Thus, the probability of getting two heads is 1 / 4 or 0.25.

Related Articles:

• Experimental Probability
• Probability Rules
• Probability and Statistics

Important Notes on Random Experiments

• A random experiment is an experiment whose outcome cannot be predicted.
• A random experiment can be performed several times under the same condition.
• The probability of a random experiment can be given by the number of favorable outcomes / total number of outcomes.

Examples on Random Experiments

Example 2: Can picking a card from a pack of cards be classified as a random experiment?

Solution: As picking a card can be done multiple times thus, this experiment can be conducted many times.

As any card can be picked up, hence, the outcome of the experiment cannot be predicted

Thus, it is a random experiment,

Answer: Picking a card from a pack of cards is a random experiment

Example 3: If there are 3 green balls, 4 red balls, and 5 pink balls in a bag then what is the probability of drawing a pink ball?

Solution: There are a total of 12 balls in the bag.

As there are 5 pink balls thus, the number of favorable outcomes is 5

P(Pink) = favorable outcomes / total number of outcomes

Answer: The probability of drawing a pink ball is 5 / 12.

go to slide go to slide go to slide

Book a Free Trial Class

FAQs on Random Experiments

What are random experiments.

Random experiments are experiments that can be performed several times and the outcome cannot be predicted beforehand.

What is the Random Experiment Sample Space and Event?

A sample space of a random experiment enlists all the possible outcomes of that experiment. However, an event is a set of possible outcomes of a random experiment that is a subset of the sample space.

What are the Two Conditions that Random Experiments Must Satisfy?

For experiments to be random experiments they must satisfy the following two conditions:

• The experiment can be arbitrarily repeated many times under the same conditions.
• The outcome of each trial of a random experiment cannot be predicted before the experiment has been performed.

What is the Formula to Find the Probability of an Outcome of a Random Experiment?

The likelihood of occurrence of any outcome of a random experiment can be calculated by the formula number of favorable outcomes / total number of outcomes.

What are the Steps to Find the Probability of a Random Experiment?

To find the probability of an outcome the steps are as follows:

• Find the total number of outcomes of the random experiment.
• Find the number of favorable outcomes.
• Divide step 2 by step 3 to determine the probability.

Can Dividing 20 by 5 Be Considered a Random Experiment?

On diving 20 by 5 the outcome will always be 4. Thus, as the outcome is predictable this cannot be classified as a random experiment.

What is a Random Variable and a Random Experiment?

A random variable is a variable that can assume all possible outcomes of a random experiment and its value changes with every trial that is performed.

Make Waves in Learning! 25% off

for World Oceans Day

Use code OCEAN25

Share this article

Table of Contents

Latest updates.

1 Million Means: 1 Million in Rupees, Lakhs and Crores

Ways To Improve Learning Outcomes: Learn Tips & Tricks

The Three States of Matter: Solids, Liquids, and Gases

Types of Motion: Introduction, Parameters, Examples

Understanding Frequency Polygon: Detailed Explanation

Uses of Silica Gel in Packaging?

Visual Learning Style for Students: Pros and Cons

Air Pollution: Know the Causes, Effects & More

Sexual Reproduction in Flowering Plants

Integers Introduction: Check Detailed Explanation

Tag cloud :.

• entrance exams
• engineering
• ssc cgl 2024
• Written By Jyoti Saxena
• Last Modified 25-01-2023

Random Experiments: Observations, Definitions, and Examples

We may engage in various  Random Experiments  in our daily lives, sometimes repeating the same behaviours and receiving the same result each time. We use words such as impossible, \(50-50\), probably, and certainly to describe the chance of a phenomenon or a result happening. 

If we toss a coin once, the outcome can be either a head or a tail. Both outcomes are equally possible. And thus, the probability of getting ahead is \(\frac{1}{2}\). Likewise, the probability of getting a tail is \(\frac{1}{2}\). 

Random Experiment in Probability

There are many situations in our daily life where we need to take a chance or risk where a particular event can be easily predicted. For example,

– My calculator will work properly during the next mathematics test.

– The sun will rise in the east tomorrow.

– There will be heavy rain two weeks from now.

– It will snow tomorrow.

– I will visit my grandmother next month.

The branch of mathematics that studies a likelihood or a chance of a phenomenon happening is known as probability. The idea of probability was developed in the \({\rm{1}}{{\rm{7}}^{{\rm{th}}}}\) century. 

An experiment is likely to have more than one possible outcome.

An event is a collection of outcomes from possible outcomes in a random experiment. Now, each of the activities mentioned above fulfils the following \(2\) conditions:

• We can repeat these activities many times under identical conditions.
• Because chance plays a part and each conclusion has the same probability of being chosen, we cannot predict the outcome of action ahead of time. As a result of the chance to play a role, an activity is performed. Those events whose outcome can not be predicted beforehand is called a random experiment.

Random Experiment Definition

A random experiment is a process in which the outcome cannot be predicted with certainty in probability. Thus, a random experiment is an experiment whose outcome cannot be predicted precisely in advance, although all possible outcomes are known. An experiment’s result is referred to as an outcome.

An event \(E\) of an experiment is a collection of outcomes. The possible outcomes (=2) when a coin is tossed, i.e., head and tail. When a dice is thrown, the possible outcomes \(=6\), i.e., \(1, 2, 3, 4, 5\), and \(6\). When a card is drawn from a deck of \(52\) cards, the event of drawing a queen consists of a queen of spades, queen of diamonds, queen of hearts, queen of clubs, and many more.

Therefore, we see that an event is part of the possible outcomes. If a random experiment has a finite number of equally likely outcomes, then the probability of an event \(E\) can be expressed as:

\(P(E) = \frac{{n(E)}}{{n(S)}}\) where \(n(E)\) is the number of outcomes favorable to the event \(E\), and \(n(S)\) is the total number of possible outcomes.

Random Experiment Examples

Let us look at some examples of random experiments in probability.

Example 1: Is drawing a card from a well-shuffled deck of cards a random experiment? Solution: While drawing a card from a well-shuffled deck of cards, the experiment can be repeated as the deck of cards can be shuffled every time before drawing a card. Also, any of the \(52\) cards can be drawn, and hence the outcome is not predictable beforehand. Therefore, this is a random experiment.

Example 3: Selecting a table from \(50\) tables without preference is a random experiment? Solution: Selecting a table from \(50\) tables is an experiment that can be repeated under identical conditions. As the selection of table is without preference, every table has an equal chance of selection, and thus the outcome is not predictable beforehand. Thus, selecting a table out of \(50\) tables is a random experiment.

Example 2: Multiplying \(2\) and \(6\) on a calculator. Solution: Although the activity of multiplying \(2\) and \(6\) can be repeated under identical conditions, the outcome is always \(12\). Hence, the activity is not a random experiment.

Random Error Experiment

Random or systematic errors account for all experimental uncertainty. Random errors are stochastic fluctuations (in either direction) in measured data caused by the measuring device’s accuracy limitations. The inability of the experimenter to take the exact measurement in precisely the same way leads to random mistakes. 

On the other hand, systematic errors are repeatable inaccuracies that always go in the same direction. Systematic errors are frequently caused by an issue that continues throughout the experiment.

It’s worth noting that systematic and random errors pertain to issues with taking measurements. 

Errors made in calculations or reading the instrument are not considered in the error analysis.

Solved Examples

Example-1: There are \(20\) seats numbered from \(1\) to \(20\) in a row in a cinema hall. If a seat is selected at random from the row, find the probability that the seat number is a) A multiple of \(3\) b) A prime number Ans: The total possible outcomes consist of \(20\) numbers which are from \(1\) to \(20\). a) The seat number should be a multiple of \(3\) if it is \(3, 6, 9, 12, 15\), or \(18\). Thus there are \(6\) multiples of \(3\) from \(1\) to \(20\). Therefore, the number of favorable outcomes \(=6\) Probability, \(P\)(the seat number is a multiple of \(3\)) \( = \frac{6}{{20}} = \frac{3}{{10}}\). b) The prime numbers between \(1\) and \(20\) are \(2, 3, 5, 7, 11, 17\), and \(19\). Thus, there are \(8\) prime numbers in between \(1\) and \(20\). Probability, \(P\)(a seat number is a prime number)\( = \frac{8}{{20}} = \frac{2}{5}\)

Example-2: A card is drawn randomly from a deck of \(52\) playing cards. The pack consists of \(4\) suits, and each suit has \(13\) cards. Find the probability that the card drawn is a) Red b) A diamond c) A Picture card Ans: There are \(13\) cards from each suit are they are Ace,\( 2, 3, 4, 5, 6, 7, 8, 9, 10\), Jack, Queen, and King.  a) Total red cards in a pack of cards \(=26\) (\(13\) hearts and \(13\) diamonds) Therefore, probability, \(P\)(Red)=\(\frac{26}{52}=\frac{1}{2}\) b) There are \(13\) diamond cards in a pack of cards. Therefore, probability, \(P\)(A diamond card)\( = \frac{{13}}{{52}} = \frac{1}{4}\) c) The picture card consists of \(4\) kings, \(4\) queens, and \(4\) jacks. Thus, the total number of picture cards \(=4+4+4=12\) Therefore, probability, \(P\)(A picture card)\( = \frac{{12}}{{52}} = \frac{3}{{13}}\)

Example-3: There are \(5\) red, \(6\) green, and \(7\) blue balls in a bag. One ball is drawn at random. What is the probability that the ball drawn is a  a) Red ball b) Green ball c) Not a green ball Ans: The total number of balls in the bag \(=5+6+7=18\) Therefore, \(P\) (Red ball)\( = \frac{5}{{18}}\) Therefore, \(P\) (Green ball)\( = \frac{6}{{18}} = \frac{1}{2}\) Out of \(18\) balls, \(12\) are not green. Therefore, \(P\) (Not a green ball)\( = \frac{5}{{18}}\).

Example-4: A fair die is rolled. Find the probability of getting  a) \(6\) b) An odd number c) A number less then \(3\) Ans: In rolling a die, there are \(6\) equally likely outcomes, i.e., \(1, 2, 3, 4, 5\), and \(6\). a) The event of getting a \(6\) consists of the one outcome \(‘6’\) Therefore, the probability of getting a \(6\), \(P\)(Getting \(6) = \frac{1}{6}\) b) There are \(3\) favorable outcomes for the event of getting an odd number. There are \(1, 3\), and \(5\). Therefore, \(P\) (Getting an odd number)\( = \frac{3}{6} = \frac{1}{2}\) The favorable outcomes of the event of getting a number less than \(3\) are \(1\) and \(2\). Therefore, \(P\)(Getting a number less than \(3) = \frac{2}{6} = \frac{1}{3}\)

Example-5: If a number of two digits is made without repetition with the digits \(1, 3, 5.\) Then what is the probability that the number formed is \(35\)? Ans: The two digits formed with the digits \(1, 3, 5\) without repetition are: \(13, 15, 31, 35, 51, 53\).  Hence, the total number of outcomes \(=6\) Out of the six numbers formed, only one number is \(35\). Therefore, the number of the outcome of the number formed being \(35=1\) Hence, probability \(= \frac{1}{6}\)

In this article, we learned about the basics of probability with examples and then learned about the random variable. We learned about random probability in detail with the help of examples.

Q.1. What is a random experiment in mathematics? Ans: A random experiment is a process in which the outcome cannot be predicted with certainty in probability. An experiment’s result is referred to as an outcome. An event \(E\) of an experiment is a collection of outcomes. When a coin is tossed, the possible outcomes \(=2\), i.e., head and tail. When a dice is thrown, the possible outcomes \(=6\).

Q.2. What do you call the collection of all possible outcomes of a random experiment? Ans: The collection of all the possible outcomes of a random experiment is known as sample space.

Q.3. Give one example which is not an example of a random experiment? Ans: A stone dropped from a rooftop is not an example of a random experiment as the outcome will always be the same, i.e., the stone will always hit the ground.

Q.4. Define probability. Ans: The branch of mathematics that studies a likelihood or a chance of a phenomenon to occur is known as probability.

Q.5. What is a random error? Ans: Random or systematic errors account for all experimental uncertainty. Random errors are stochastic fluctuations (in either direction) in measured data caused by the measuring device’s accuracy limitations. The inability of the experimenter to take the exact measurement, in the same way, leads to random mistakes.

We hope you find this detailed article on random experiments helpful. If you have any doubts or queries regarding this topic, feel to ask us in the comment section. Happy learning!

Related Articles

1 Million Means: 1 million in numerical is represented as 10,00,000. The Indian equivalent of a million is ten lakh rupees. It is not a...

Ways To Improve Learning Outcomes: With the development of technology, students may now rely on strategies to enhance learning outcomes. No matter how knowledgeable a...

The Three States of Matter: Anything with mass and occupied space is called ‘Matter’. Matters of different kinds surround us. There are some we can...

Motion is the change of a body's position or orientation over time. The motion of humans and animals illustrates how everything in the cosmos is...

Understanding Frequency Polygon: Students who are struggling with understanding Frequency Polygon can check out the details here. A graphical representation of data distribution helps understand...

When you receive your order of clothes or leather shoes or silver jewellery from any online shoppe, you must have noticed a small packet containing...

Visual Learning Style: We as humans possess the power to remember those which we have caught visually in our memory and that too for a...

Air Pollution: In the past, the air we inhaled was pure and clean. But as industrialisation grows and the number of harmful chemicals in the...

In biology, flowering plants are known by the name angiosperms. Male and female reproductive organs can be found in the same plant in flowering plants....

Integers Introduction: To score well in the exam, students must check out the Integers introduction and understand them thoroughly. The collection of negative numbers and whole...

Human Respiratory System – Detailed Explanation

Human Respiratory System: Students preparing for the NEET and Biology-related exams must have an idea about the human respiratory system. It is a network of tissues...

Place Value of Numbers: Detailed Explanation

Place Value of Numbers: Students must understand the concept of the place value of numbers to score high in the exam. In mathematics, place value...

The Leaf: Types, Structures, Parts

The Leaf: Students who want to understand everything about the leaf can check out the detailed explanation provided by Embibe experts. Plants have a crucial role...

Factors Affecting Respiration: Definition, Diagrams with Examples

In plants, respiration can be regarded as the reversal of the photosynthetic process. Like photosynthesis, respiration involves gas exchange with the environment. Unlike photosynthesis, respiration...

General Terms Related to Spherical Mirrors

General terms related to spherical mirrors: A mirror with the shape of a portion cut out of a spherical surface or substance is known as a...

Number System: Types, Conversion and Properties

Number System: Numbers are highly significant and play an essential role in Mathematics that will come up in further classes. In lower grades, we learned how...

Types of Respiration

Every living organism has to "breathe" to survive. The process by which the living organisms use their food to get energy is called respiration. It...

Animal Cell: Definition, Diagram, Types of Animal Cells

Animal Cell: An animal cell is a eukaryotic cell with membrane-bound cell organelles without a cell wall. We all know that the cell is the fundamental...

Conversion of Percentages: Conversion Method & Examples

Conversion of Percentages: To differentiate and explain the size of quantities, the terms fractions and percent are used interchangeably. Some may find it difficult to...

Arc of a Circle: Definition, Properties, and Examples

Arc of a circle: A circle is the set of all points in the plane that are a fixed distance called the radius from a fixed point...

Ammonia (NH3): Preparation, Structure, Properties and Uses

Ammonia, a colourless gas with a distinct odour, is a chemical building block and a significant component in producing many everyday items. It is found...

CGPA to Percentage: Calculator for Conversion, Formula, & More

CGPA to Percentage: The average grade point of a student is calculated using their cumulative grades across all subjects, omitting any supplemental coursework. Many colleges,...

Uses of Ether – Properties, Nomenclature, Uses, Disadvantages

Uses of Ether:  Ether is an organic compound containing an oxygen atom and an ether group connected to two alkyl/aryl groups. It is formed by the...

General and Middle Terms: Definitions, Formula, Independent Term, Examples

General and Middle terms: The binomial theorem helps us find the power of a binomial without going through the tedious multiplication process. Further, the use...

Mutually Exclusive Events: Definition, Formulas, Solved Examples

Mutually Exclusive Events: In the theory of probability, two events are said to be mutually exclusive events if they cannot occur simultaneously or at the...

Geometry: Definition, Shapes, Structure, Examples

Geometry is a branch of mathematics that is largely concerned with the forms and sizes of objects, their relative positions, and the qualities of space....

Bohr’s Model of Hydrogen Atom: Expressions for Radius, Energy

Rutherford’s Atom Model was undoubtedly a breakthrough in atomic studies. However, it was not wholly correct. The great Danish physicist Niels Bohr (1885–1962) made immediate...

39 Insightful Publications

Embibe Is A Global Innovator

Innovator Of The Year Education Forever

Interpretable And Explainable AI

Revolutionizing Education Forever

Best AI Platform For Education

Enabling Teachers Everywhere

Decoding Performance

Leading AI Powered Learning Solution Provider

Auto Generation Of Tests

Disrupting Education In India

Problem Sequencing Using DKT

Help Students Ace India's Toughest Exams

Best Education AI Platform

Unlocking AI Through Saas

Fixing Student’s Behaviour With Data Analytics

Leveraging Intelligence To Deliver Results

Brave New World Of Applied AI

You Can Score Higher

Harnessing AI In Education

Personalized Ed-tech With AI

Exciting AI Platform, Personalizing Education

Disruptor Award For Maximum Business Impact

Top 20 AI Influencers In India

Proud Owner Of 9 Patents

Innovation in AR/VR/MR

Best Animated Frames Award 2024

Trending Searches

Previous year question papers, sample papers.

Unleash Your True Potential With Personalised Learning on EMBIBE

Ace Your Exam With Personalised Learning on EMBIBE

Enter mobile number.

By signing up, you agree to our Privacy Policy and Terms & Conditions

• FOR INSTRUCTOR
• FOR INSTRUCTORS

1.3.1 Random Experiments

• Random experiment: toss a coin; sample space: $S=\{heads, tails\}$ or as we usually write it, $\{H,T\}$.
• Random experiment: roll a die; sample space: $S=\{1, 2, 3, 4, 5, 6\}$.
• Random experiment: observe the number of iPhones sold by an Apple store in Boston in $2015$; sample space: $S=\{0, 1, 2, 3, \cdots \}$.
• Random experiment: observe the number of goals in a soccer match; sample space: $S=\{0, 1, 2, 3, \cdots \}$.

When we repeat a random experiment several times, we call each one of them a trial . Thus, a trial is a particular performance of a random experiment. In the example of tossing a coin, each trial will result in either heads or tails. Note that the sample space is defined based on how you define your random experiment. For example,

Example We toss a coin three times and observe the sequence of heads/tails. The sample space here may be defined as $$S = \{(H,H,H), (H,H,T), (H,T,H), (T,H,H), (H,T,T),(T,H,T),(T,T,H),(T,T,T)\}.$$

Our goal is to assign probability to certain events . For example, suppose that we would like to know the probability that the outcome of rolling a fair die is an even number. In this case, our event is the set $E=\{2, 4, 6\}$. If the result of our random experiment belongs to the set $E$, we say that the event $E$ has occurred. Thus an event is a collection of possible outcomes. In other words, an event is a subset of the sample space to which we assign a probability. Although we have not yet discussed how to find the probability of an event, you might be able to guess that the probability of $\{2, 4, 6 \}$ is $50$ percent which is the same as $\frac{1}{2}$ in the probability theory convention.

Outcome: A result of a random experiment. Sample Space: The set of all possible outcomes. Event: A subset of the sample space.

Union and Intersection: If $A$ and $B$ are events, then $A \cup B$ and $A \cap B$ are also events. By remembering the definition of union and intersection, we observe that $A \cup B$ occurs if $A$ or $B$ occur. Similarly, $A \cap B$ occurs if both $A$ and $B$ occur. Similarly, if $A_1, A_2,\cdots, A_n$ are events, then the event $A_1 \cup A_2 \cup A_3 \cdots \cup A_n$ occurs if at least one of $A_1, A_2,\cdots, A_n$ occurs. The event $A_1 \cap A_2 \cap A_3 \cdots \cap A_n$ occurs if all of $A_1, A_2,\cdots, A_n$ occur. It can be helpful to remember that the key words "or" and "at least" correspond to unions and the key words "and" and "all of" correspond to intersections.

MA121: Introduction to Statistics

Basic Concepts of Probability

Read this section about basic concepts of probability, including spaces, and events. This section discusses set operations using Venn diagrams, including complements, intersections, and unions. Finally, it introduces conditional probability and talks about independent events.

LEARNING OBJECTIVES

• To learn the concept of the sample space associated with a random experiment.
• To learn the concept of an event associated with a random experiment.
• To learn the concept of the probability of an event.

Sample Spaces and Events

Rolling an ordinary six-sided die is a familiar example of a random experiment, an action for which all possible outcomes can be listed, but for which the actual outcome on any given trial of the experiment cannot be predicted with certainty. In such a situation we wish to assign to each outcome, such as rolling a two, a number, called the probability of the outcome, that indicates how likely it is that the outcome will occur. Similarly, we would like to assign a probability to any event, or collection of outcomes, such as rolling an even number, which indicates how likely it is that the event will occur if the experiment is performed. This section provides a framework for discussing probability problems, using the terms just mentioned.

A random experiment is a mechanism that produces a definite outcome that cannot be predicted with certainty. The sample space associated with a random experiment is the set of all possible outcomes. An event is a subset of the sample space .

Construct a sample space for the experiment that consists of tossing a single coin.

Construct a sample space for the experiment that consists of rolling a single die. Find the events that correspond to the phrases "an even number is rolled" and "a number greater than two is rolled".

Figure 3.1 Venn Diagrams for Two Sample Spaces

A random experiment consists of tossing two coins.

a. Construct a sample space for the situation that the coins are indistinguishable, such as two brand new pennies.

b. Construct a sample space for the situation that the coins are distinguishable, such as one a penny and the other a nickel.

A device that can be helpful in identifying all possible outcomes of a random experiment, particularly one that can be viewed as proceeding in stages, is what is called a tree diagram. It is described in the following example.

Construct a sample space that describes all three-child families according to the genders of the children with respect to birth order.

Tree Diagram For Three-Child Families

The line segments are called branches of the tree. The right ending point of each branch is called a node. The nodes on the extreme right are the final nodes ; to each one there corresponds an outcome, as shown in the figure.

From the tree it is easy to read off the eight outcomes of the experiment, so the sample space is, reading from the top to the bottom of the final nodes in the tree,

Random Experiments

Random experiments are also known as observations . The word ‘Probability’ is used very often in our daily life; such as ‘probably he is an honest man’, ‘what is the probability of a double head in a throw of a pair of coin?’, ‘probably it will rain in the evening’ and so on.

These days, an attempt towards a theory of probability is extensively used in various fields of what we are interested in. The instant answer is obviously an event. At first we will discuss about the precise meaning of the term ‘ event ’ and how it is used in our mathematical theory. Now our next step will help us to get the idea of  what are random experiments or observations .

Definition of random experiment:

An experiment for which we know the set of all different results but it is not possible to predict which one of the set will occur at any particular execution of the experiment is called a  random experiment .

For example: tossing a fair coin, casting an unbiased die and drawing a card from a pack of 52 cards.

Let us take the experiment of tossing a coin . If we toss a coin then we get two possible outcomes either a ‘head’ (H) or a ‘tail’ (T) , and it is impossible to predict whether the result of a toss will be a ‘head’ or ‘tail’.

Let us consider a similar experiment rolling a die from a box. It we a die then there are only six possible outcomes. The faces of the die are marked as 1, 2, 3, 4, 5 and 6 and these are the only possible outcomes. But here also the outcome of a particular throw is completely unpredictable.

Again similarly when we are concerned about the measurement of a chemical quantity with the required instrument the outcome of the experiment does not exactly give the true value of the quantity but a value close to it due to that are called experimental errors. If repeated observations are taken the measured values are not again the same to the previous one but it will fluctuate in an unpredictable manner. Here we can take, at least for the theoretical considerations that the possible outcomes comprise all the real numbers, but the number given by a single measurement can’t be exactly predicted. In our mathematical theory we will only consider the experiments or observations, for which we know a priori the set of all different possible outcomes, such that it is impossible to predict which particular outcome will occur at any particular performance of the experiment, are called random experiments. As such, if a random experiment is repeated under identical conditions the outcomes or results may vary or fluctuate at random.

• Probability

Experimental Probability

Events in Probability

Empirical Probability

Coin Toss Probability

Probability of Tossing Two Coins

Probability of Tossing Three Coins

Complimentary Events

Mutually Exclusive Events

Mutually Non-Exclusive Events

Conditional Probability

Theoretical Probability

Odds and Probability

Playing Cards Probability

Probability and Playing Cards

Probability for Rolling Two Dice

Solved Probability Problems

Probability for Rolling Three Dice

• 9th Grade Math

From Random Experiments to HOME PAGE

Didn't find what you were looking for? Or want to know more information about Math Only Math . Use this Google Search to find what you need.

New! Comments

Share this page: What’s this?

• Preschool Activities
• Kindergarten Math
• 1st Grade Math
• 2nd Grade Math
• 3rd Grade Math
• 4th Grade Math
• 5th Grade Math
• 6th Grade Math
• 7th Grade Math
• 8th Grade Math
• 10th Grade Math
• 11 & 12 Grade Math
• Concepts of Sets
• Boolean Algebra
• Math Coloring Pages
• Multiplication Table
• Cool Maths Games
• Math Flash Cards
• Online Math Quiz
• Math Puzzles
• Binary System
• Math Dictionary
• Conversion Chart
• Homework Sheets
• Math Problem Ans
• Free Math Answers
• Printable Math Sheet
• Funny Math Answers
• Employment Test
• Math Patterns
• Link Partners
• Privacy Policy

Recent Articles

Construction of bar graphs | examples on construction of column graph.

Aug 29, 24 02:26 AM

5th Grade Bar Graph | Definition | Interpret Bar Graphs|Free Worksheet

Aug 29, 24 02:11 AM

Worksheet on Pictographs | Picture Graph Worksheets | Pictograph Works

Aug 29, 24 12:38 AM

Worksheet on Data Handling | Questions on Handling Data |Grouping Data

Aug 28, 24 10:42 AM

Pictographs | Pictorial Representation | Pictorial Symbols | Symbols

Aug 27, 24 05:56 PM

© and ™ math-only-math.com. All Rights Reserved. 2010 - 2024.

Have a language expert improve your writing

Run a free plagiarism check in 10 minutes, generate accurate citations for free.

• Knowledge Base

Methodology

• Random Assignment in Experiments | Introduction & Examples

Random Assignment in Experiments | Introduction & Examples

Published on March 8, 2021 by Pritha Bhandari . Revised on June 22, 2023.

In experimental research, random assignment is a way of placing participants from your sample into different treatment groups using randomization.

With simple random assignment, every member of the sample has a known or equal chance of being placed in a control group or an experimental group. Studies that use simple random assignment are also called completely randomized designs .

Random assignment is a key part of experimental design . It helps you ensure that all groups are comparable at the start of a study: any differences between them are due to random factors, not research biases like sampling bias or selection bias .

Table of contents

Why does random assignment matter, random sampling vs random assignment, how do you use random assignment, when is random assignment not used, other interesting articles, frequently asked questions about random assignment.

Random assignment is an important part of control in experimental research, because it helps strengthen the internal validity of an experiment and avoid biases.

In experiments, researchers manipulate an independent variable to assess its effect on a dependent variable, while controlling for other variables. To do so, they often use different levels of an independent variable for different groups of participants.

This is called a between-groups or independent measures design.

You use three groups of participants that are each given a different level of the independent variable:

• a control group that’s given a placebo (no dosage, to control for a placebo effect ),
• an experimental group that’s given a low dosage,
• a second experimental group that’s given a high dosage.

Random assignment to helps you make sure that the treatment groups don’t differ in systematic ways at the start of the experiment, as this can seriously affect (and even invalidate) your work.

If you don’t use random assignment, you may not be able to rule out alternative explanations for your results.

• participants recruited from cafes are placed in the control group ,
• participants recruited from local community centers are placed in the low dosage experimental group,
• participants recruited from gyms are placed in the high dosage group.

With this type of assignment, it’s hard to tell whether the participant characteristics are the same across all groups at the start of the study. Gym-users may tend to engage in more healthy behaviors than people who frequent cafes or community centers, and this would introduce a healthy user bias in your study.

Although random assignment helps even out baseline differences between groups, it doesn’t always make them completely equivalent. There may still be extraneous variables that differ between groups, and there will always be some group differences that arise from chance.

Most of the time, the random variation between groups is low, and, therefore, it’s acceptable for further analysis. This is especially true when you have a large sample. In general, you should always use random assignment in experiments when it is ethically possible and makes sense for your study topic.

Receive feedback on language, structure, and formatting

Professional editors proofread and edit your paper by focusing on:

• Academic style
• Vague sentences
• Style consistency

See an example

Random sampling and random assignment are both important concepts in research, but it’s important to understand the difference between them.

Random sampling (also called probability sampling or random selection) is a way of selecting members of a population to be included in your study. In contrast, random assignment is a way of sorting the sample participants into control and experimental groups.

While random sampling is used in many types of studies, random assignment is only used in between-subjects experimental designs.

Some studies use both random sampling and random assignment, while others use only one or the other.

Random sampling enhances the external validity or generalizability of your results, because it helps ensure that your sample is unbiased and representative of the whole population. This allows you to make stronger statistical inferences .

You use a simple random sample to collect data. Because you have access to the whole population (all employees), you can assign all 8000 employees a number and use a random number generator to select 300 employees. These 300 employees are your full sample.

Random assignment enhances the internal validity of the study, because it ensures that there are no systematic differences between the participants in each group. This helps you conclude that the outcomes can be attributed to the independent variable .

• a control group that receives no intervention.
• an experimental group that has a remote team-building intervention every week for a month.

You use random assignment to place participants into the control or experimental group. To do so, you take your list of participants and assign each participant a number. Again, you use a random number generator to place each participant in one of the two groups.

To use simple random assignment, you start by giving every member of the sample a unique number. Then, you can use computer programs or manual methods to randomly assign each participant to a group.

• Random number generator: Use a computer program to generate random numbers from the list for each group.
• Lottery method: Place all numbers individually in a hat or a bucket, and draw numbers at random for each group.
• Flip a coin: When you only have two groups, for each number on the list, flip a coin to decide if they’ll be in the control or the experimental group.
• Use a dice: When you have three groups, for each number on the list, roll a dice to decide which of the groups they will be in. For example, assume that rolling 1 or 2 lands them in a control group; 3 or 4 in an experimental group; and 5 or 6 in a second control or experimental group.

This type of random assignment is the most powerful method of placing participants in conditions, because each individual has an equal chance of being placed in any one of your treatment groups.

Random assignment in block designs

In more complicated experimental designs, random assignment is only used after participants are grouped into blocks based on some characteristic (e.g., test score or demographic variable). These groupings mean that you need a larger sample to achieve high statistical power .

For example, a randomized block design involves placing participants into blocks based on a shared characteristic (e.g., college students versus graduates), and then using random assignment within each block to assign participants to every treatment condition. This helps you assess whether the characteristic affects the outcomes of your treatment.

In an experimental matched design , you use blocking and then match up individual participants from each block based on specific characteristics. Within each matched pair or group, you randomly assign each participant to one of the conditions in the experiment and compare their outcomes.

Sometimes, it’s not relevant or ethical to use simple random assignment, so groups are assigned in a different way.

When comparing different groups

Sometimes, differences between participants are the main focus of a study, for example, when comparing men and women or people with and without health conditions. Participants are not randomly assigned to different groups, but instead assigned based on their characteristics.

In this type of study, the characteristic of interest (e.g., gender) is an independent variable, and the groups differ based on the different levels (e.g., men, women, etc.). All participants are tested the same way, and then their group-level outcomes are compared.

When it’s not ethically permissible

When studying unhealthy or dangerous behaviors, it’s not possible to use random assignment. For example, if you’re studying heavy drinkers and social drinkers, it’s unethical to randomly assign participants to one of the two groups and ask them to drink large amounts of alcohol for your experiment.

When you can’t assign participants to groups, you can also conduct a quasi-experimental study . In a quasi-experiment, you study the outcomes of pre-existing groups who receive treatments that you may not have any control over (e.g., heavy drinkers and social drinkers). These groups aren’t randomly assigned, but may be considered comparable when some other variables (e.g., age or socioeconomic status) are controlled for.

Prevent plagiarism. Run a free check.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Student’s  t -distribution
• Normal distribution
• Null and Alternative Hypotheses
• Chi square tests
• Confidence interval
• Quartiles & Quantiles
• Cluster sampling
• Stratified sampling
• Data cleansing
• Reproducibility vs Replicability
• Peer review
• Prospective cohort study

Research bias

• Implicit bias
• Cognitive bias
• Placebo effect
• Hawthorne effect
• Hindsight bias
• Affect heuristic
• Social desirability bias

In experimental research, random assignment is a way of placing participants from your sample into different groups using randomization. With this method, every member of the sample has a known or equal chance of being placed in a control group or an experimental group.

Random selection, or random sampling , is a way of selecting members of a population for your study’s sample.

In contrast, random assignment is a way of sorting the sample into control and experimental groups.

Random sampling enhances the external validity or generalizability of your results, while random assignment improves the internal validity of your study.

Random assignment is used in experiments with a between-groups or independent measures design. In this research design, there’s usually a control group and one or more experimental groups. Random assignment helps ensure that the groups are comparable.

In general, you should always use random assignment in this type of experimental design when it is ethically possible and makes sense for your study topic.

To implement random assignment , assign a unique number to every member of your study’s sample .

Then, you can use a random number generator or a lottery method to randomly assign each number to a control or experimental group. You can also do so manually, by flipping a coin or rolling a dice to randomly assign participants to groups.

Cite this Scribbr article

If you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator.

Bhandari, P. (2023, June 22). Random Assignment in Experiments | Introduction & Examples. Scribbr. Retrieved August 26, 2024, from https://www.scribbr.com/methodology/random-assignment/

Is this article helpful?

Pritha Bhandari

Other students also liked, guide to experimental design | overview, steps, & examples, confounding variables | definition, examples & controls, control groups and treatment groups | uses & examples, what is your plagiarism score.

3.1 Sample Spaces, Events, and Their Probabilities

Learning objectives.

• To learn the concept of the sample space associated with a random experiment.
• To learn the concept of an event associated with a random experiment.
• To learn the concept of the probability of an event.

Sample Spaces and Events

Rolling an ordinary six-sided die is a familiar example of a random experiment , an action for which all possible outcomes can be listed, but for which the actual outcome on any given trial of the experiment cannot be predicted with certainty. In such a situation we wish to assign to each outcome, such as rolling a two, a number, called the probability of the outcome, that indicates how likely it is that the outcome will occur. Similarly, we would like to assign a probability to any event , or collection of outcomes, such as rolling an even number, which indicates how likely it is that the event will occur if the experiment is performed. This section provides a framework for discussing probability problems, using the terms just mentioned.

A random experiment is a mechanism that produces a definite outcome that cannot be predicted with certainty. The sample space The set of all possible outcomes of a random experiment. associated with a random experiment is the set of all possible outcomes. An event Any set of outcomes. is a subset of the sample space.

An event E is said to occur on a particular trial of the experiment if the outcome observed is an element of the set E .

Construct a sample space for the experiment that consists of tossing a single coin.

The outcomes could be labeled h for heads and t for tails. Then the sample space is the set S = { h , t } .

Construct a sample space for the experiment that consists of rolling a single die. Find the events that correspond to the phrases “an even number is rolled” and “a number greater than two is rolled.”

The outcomes could be labeled according to the number of dots on the top face of the die. Then the sample space is the set S = { 1,2,3,4,5,6 } .

The outcomes that are even are 2, 4, and 6, so the event that corresponds to the phrase “an even number is rolled” is the set {2,4,6}, which it is natural to denote by the letter E . We write E = { 2,4,6 } .

Similarly the event that corresponds to the phrase “a number greater than two is rolled” is the set T = { 3,4,5,6 } , which we have denoted T .

A graphical representation of a sample space and events is a Venn diagram , as shown in Figure 3.1 "Venn Diagrams for Two Sample Spaces" for Note 3.6 "Example 1" and Note 3.7 "Example 2" . In general the sample space S is represented by a rectangle, outcomes by points within the rectangle, and events by ovals that enclose the outcomes that compose them.

Figure 3.1 Venn Diagrams for Two Sample Spaces

A random experiment consists of tossing two coins.

• Construct a sample space for the situation that the coins are indistinguishable, such as two brand new pennies.
• Construct a sample space for the situation that the coins are distinguishable, such as one a penny and the other a nickel.
• After the coins are tossed one sees either two heads, which could be labeled 2 h , two tails, which could be labeled 2 t , or coins that differ, which could be labeled d . Thus a sample space is S = { 2 h , 2 t , d } .
• Since we can tell the coins apart, there are now two ways for the coins to differ: the penny heads and the nickel tails, or the penny tails and the nickel heads. We can label each outcome as a pair of letters, the first of which indicates how the penny landed and the second of which indicates how the nickel landed. A sample space is then S ′ = { h h , h t , t h , t t } .

A device that can be helpful in identifying all possible outcomes of a random experiment, particularly one that can be viewed as proceeding in stages, is what is called a tree diagram . It is described in the following example.

Construct a sample space that describes all three-child families according to the genders of the children with respect to birth order.

Two of the outcomes are “two boys then a girl,” which we might denote b b g , and “a girl then two boys,” which we would denote g b b . Clearly there are many outcomes, and when we try to list all of them it could be difficult to be sure that we have found them all unless we proceed systematically. The tree diagram shown in Figure 3.2 "Tree Diagram For Three-Child Families" , gives a systematic approach.

Figure 3.2 Tree Diagram For Three-Child Families

The diagram was constructed as follows. There are two possibilities for the first child, boy or girl, so we draw two line segments coming out of a starting point, one ending in a b for “boy” and the other ending in a g for “girl.” For each of these two possibilities for the first child there are two possibilities for the second child, “boy” or “girl,” so from each of the b and g we draw two line segments, one segment ending in a b and one in a g . For each of the four ending points now in the diagram there are two possibilities for the third child, so we repeat the process once more.

The line segments are called branches of the tree. The right ending point of each branch is called a node . The nodes on the extreme right are the final nodes ; to each one there corresponds an outcome, as shown in the figure.

From the tree it is easy to read off the eight outcomes of the experiment, so the sample space is, reading from the top to the bottom of the final nodes in the tree,

Probability

The probability of an outcome A number that measures the likelihood of the outcome. e in a sample space S is a number p between 0 and 1 that measures the likelihood that e will occur on a single trial of the corresponding random experiment. The value p = 0 corresponds to the outcome e being impossible and the value p = 1 corresponds to the outcome e being certain.

The probability of an event A number that measures the likelihood of the event. A is the sum of the probabilities of the individual outcomes of which it is composed. It is denoted P ( A ) .

The following formula expresses the content of the definition of the probability of an event:

If an event E is E = { e 1 , e 2 , … , e k } , then

Figure 3.3 "Sample Spaces and Probability" graphically illustrates the definitions.

Figure 3.3 Sample Spaces and Probability

Since the whole sample space S is an event that is certain to occur, the sum of the probabilities of all the outcomes must be the number 1.

In ordinary language probabilities are frequently expressed as percentages. For example, we would say that there is a 70% chance of rain tomorrow, meaning that the probability of rain is 0.70. We will use this practice here, but in all the computational formulas that follow we will use the form 0.70 and not 70%.

A coin is called “balanced” or “fair” if each side is equally likely to land up. Assign a probability to each outcome in the sample space for the experiment that consists of tossing a single fair coin.

With the outcomes labeled h for heads and t for tails, the sample space is the set S = { h , t } . Since the outcomes have the same probabilities, which must add up to 1, each outcome is assigned probability 1/2.

A die is called “balanced” or “fair” if each side is equally likely to land on top. Assign a probability to each outcome in the sample space for the experiment that consists of tossing a single fair die. Find the probabilities of the events E : “an even number is rolled” and T : “a number greater than two is rolled.”

With outcomes labeled according to the number of dots on the top face of the die, the sample space is the set S = { 1,2,3,4,5,6 } . Since there are six equally likely outcomes, which must add up to 1, each is assigned probability 1/6.

Since E = { 2,4,6 } , P ( E ) = 1 ∕ 6 + 1 ∕ 6 + 1 ∕ 6 = 3 ∕ 6 = 1 ∕ 2 .

Since T = { 3,4,5,6 } , P ( T ) = 4 ∕ 6 = 2 ∕ 3 .

Two fair coins are tossed. Find the probability that the coins match, i.e., either both land heads or both land tails.

In Note 3.8 "Example 3" we constructed the sample space S = { 2 h , 2 t , d } for the situation in which the coins are identical and the sample space S ′ = { h h , h t , t h , t t } for the situation in which the two coins can be told apart.

The theory of probability does not tell us how to assign probabilities to the outcomes, only what to do with them once they are assigned. Specifically, using sample space S , matching coins is the event M = { 2 h , 2 t } , which has probability P ( 2 h ) + P ( 2 t ) . Using sample space S ′ , matching coins is the event M ′ = { h h , t t } , which has probability P ( h h ) + P ( t t ) . In the physical world it should make no difference whether the coins are identical or not, and so we would like to assign probabilities to the outcomes so that the numbers P ( M ) and P ( M ′ ) are the same and best match what we observe when actual physical experiments are performed with coins that seem to be fair. Actual experience suggests that the outcomes in S ′ are equally likely, so we assign to each probability 1∕4, and then

Similarly, from experience appropriate choices for the outcomes in S are:

which give the same final answer

The previous three examples illustrate how probabilities can be computed simply by counting when the sample space consists of a finite number of equally likely outcomes. In some situations the individual outcomes of any sample space that represents the experiment are unavoidably unequally likely, in which case probabilities cannot be computed merely by counting, but the computational formula given in the definition of the probability of an event must be used.

The breakdown of the student body in a local high school according to race and ethnicity is 51% white, 27% black, 11% Hispanic, 6% Asian, and 5% for all others. A student is randomly selected from this high school. (To select “randomly” means that every student has the same chance of being selected.) Find the probabilities of the following events:

• B : the student is black,
• M : the student is minority (that is, not white),
• N : the student is not black.

The experiment is the action of randomly selecting a student from the student population of the high school. An obvious sample space is S = { w , b , h , a , o } . Since 51% of the students are white and all students have the same chance of being selected, P ( w ) = 0.51 , and similarly for the other outcomes. This information is summarized in the following table:

• Since B = { b } , P ( B ) = P ( b ) = 0.27 .
• Since M = { b , h , a , o } , P ( M ) = P ( b ) + P ( h ) + P ( a ) + P ( o ) = 0.27 + 0.11 + 0.06 + 0.05 = 0.49
• Since N = { w , h , a , o } , P ( N ) = P ( w ) + P ( h ) + P ( a ) + P ( o ) = 0.51 + 0.11 + 0.06 + 0.05 = 0.73

The student body in the high school considered in Note 3.18 "Example 8" may be broken down into ten categories as follows: 25% white male, 26% white female, 12% black male, 15% black female, 6% Hispanic male, 5% Hispanic female, 3% Asian male, 3% Asian female, 1% male of other minorities combined, and 4% female of other minorities combined. A student is randomly selected from this high school. Find the probabilities of the following events:

• M F : the student is minority female,
• F N : the student is female and is not black.

Now the sample space is S = { w m , b m , h m , a m , o m , w f , b f , h f , a f , o f } . The information given in the example can be summarized in the following table, called a two-way contingency table :

• Since B = { b m , b f } , P ( B ) = P ( b m ) + P ( b f ) = 0.12 + 0.15 = 0.27 .
• Since M F = { b f , h f , a f , o f } , P ( M ) = P ( b f ) + P ( h f ) + P ( a f ) + P ( o f ) = 0.15 + 0.05 + 0.03 + 0.04 = 0.27
• Since F N = { w f , h f , a f , o f } , P ( F N ) = P ( w f ) + P ( h f ) + P ( a f ) + P ( o f ) = 0.26 + 0.05 + 0.03 + 0.04 = 0.38

Key Takeaways

• The sample space of a random experiment is the collection of all possible outcomes.
• An event associated with a random experiment is a subset of the sample space.
• The probability of any outcome is a number between 0 and 1. The probabilities of all the outcomes add up to 1.
• The probability of any event A is the sum of the probabilities of the outcomes in A .

A box contains 10 white and 10 black marbles. Construct a sample space for the experiment of randomly drawing out, with replacement, two marbles in succession and noting the color each time. (To draw “with replacement” means that the first marble is put back before the second marble is drawn.)

A box contains 16 white and 16 black marbles. Construct a sample space for the experiment of randomly drawing out, with replacement, three marbles in succession and noting the color each time. (To draw “with replacement” means that each marble is put back before the next marble is drawn.)

A box contains 8 red, 8 yellow, and 8 green marbles. Construct a sample space for the experiment of randomly drawing out, with replacement, two marbles in succession and noting the color each time.

A box contains 6 red, 6 yellow, and 6 green marbles. Construct a sample space for the experiment of randomly drawing out, with replacement, three marbles in succession and noting the color each time.

In the situation of Exercise 1, list the outcomes that comprise each of the following events.

• At least one marble of each color is drawn.
• No white marble is drawn.

In the situation of Exercise 2, list the outcomes that comprise each of the following events.

• More black than white marbles are drawn.

In the situation of Exercise 3, list the outcomes that comprise each of the following events.

• No yellow marble is drawn.
• The two marbles drawn have the same color.

In the situation of Exercise 4, list the outcomes that comprise each of the following events.

• The three marbles drawn have the same color.

Assuming that each outcome is equally likely, find the probability of each event in Exercise 5.

Assuming that each outcome is equally likely, find the probability of each event in Exercise 6.

Assuming that each outcome is equally likely, find the probability of each event in Exercise 7.

Assuming that each outcome is equally likely, find the probability of each event in Exercise 8.

A sample space is S = { a , b , c , d , e } . Identify two events as U = { a , b , d } and V = { b , c , d } . Suppose P ( a ) and P ( b ) are each 0.2 and P ( c ) and P ( d ) are each 0.1.

• Determine what P ( e ) must be.
• Find P ( U ) .
• Find P ( V ) .

A sample space is S = { u , v , w , x } . Identify two events as A = { v , w } and B = { u , w , x } . Suppose P ( u ) = 0.22 , P ( w ) = 0.36 , and P ( x ) = 0.27 .

• Determine what P ( v ) must be.
• Find P ( A ) .
• Find P ( B ) .

A sample space is S = { m , n , q , r , s } . Identify two events as U = { m , q , s } and V = { n , q , r } . The probabilities of some of the outcomes are given by the following table:

• Determine what P ( q ) must be.

A sample space is S = { d , e , f , g , h } . Identify two events as M = { e , f , g , h } and N = { d , g } . The probabilities of some of the outcomes are given by the following table:

• Determine what P ( g ) must be.
• Find P ( M ) .
• Find P ( N ) .

Applications

The sample space that describes all three-child families according to the genders of the children with respect to birth order was constructed in Note 3.9 "Example 4" . Identify the outcomes that comprise each of the following events in the experiment of selecting a three-child family at random.

• At least one child is a girl.
• At most one child is a girl.
• All of the children are girls.
• Exactly two of the children are girls.
• The first born is a girl.

The sample space that describes three tosses of a coin is the same as the one constructed in Note 3.9 "Example 4" with “boy” replaced by “heads” and “girl” replaced by “tails.” Identify the outcomes that comprise each of the following events in the experiment of tossing a coin three times.

• The coin lands heads more often than tails.
• The coin lands heads the same number of times as it lands tails.
• The coin lands heads at least twice.
• The coin lands heads on the last toss.

Assuming that the outcomes are equally likely, find the probability of each event in Exercise 17.

Assuming that the outcomes are equally likely, find the probability of each event in Exercise 18.

Additional Exercises

The following two-way contingency table gives the breakdown of the population in a particular locale according to age and tobacco usage:

A person is selected at random. Find the probability of each of the following events.

• The person is a smoker.
• The person is under 30.
• The person is a smoker who is under 30.

The following two-way contingency table gives the breakdown of the population in a particular locale according to party affiliation ( A , B , C , or None ) and opinion on a bond issue:

• The person is affiliated with party B .
• The person is affiliated with some party.
• The person is in favor of the bond issue.
• The person has no party affiliation and is undecided about the bond issue.

The following two-way contingency table gives the breakdown of the population of married or previously married women beyond child-bearing age in a particular locale according to age at first marriage and number of children:

A woman is selected at random. Find the probability of each of the following events.

• The woman was in her twenties at her first marriage.
• The woman was 20 or older at her first marriage.
• The woman had no children.
• The woman was in her twenties at her first marriage and had at least three children.

The following two-way contingency table gives the breakdown of the population of adults in a particular locale according to highest level of education and whether or not the individual regularly takes dietary supplements:

An adult is selected at random. Find the probability of each of the following events.

• The person has a high school diploma and takes dietary supplements regularly.
• The person has an undergraduate degree and takes dietary supplements regularly.
• The person takes dietary supplements regularly.
• The person does not take dietary supplements regularly.

Large Data Set Exercises

Note: These data sets are missing, but the questions are provided here for reference.

Large Data Sets 4 and 4A record the results of 500 tosses of a coin. Find the relative frequency of each outcome 1, 2, 3, 4, 5, and 6. Does the coin appear to be “balanced” or “fair”?

Large Data Sets 6, 6A, and 6B record results of a random survey of 200 voters in each of two regions, in which they were asked to express whether they prefer Candidate A for a U.S. Senate seat or prefer some other candidate.

• Find the probability that a randomly selected voter among these 400 prefers Candidate A .
• Find the probability that a randomly selected voter among the 200 who live in Region 1 prefers Candidate A (separately recorded in Large Data Set 6A).
• Find the probability that a randomly selected voter among the 200 who live in Region 2 prefers Candidate A (separately recorded in Large Data Set 6B).

S = { b b , b w , w b , w w }

S = { r r , r y , r g , y r , y y , y g , g r , g y , g g }

• { b w , w b }
• { r r , r g , g r , g g }
• { r r , y y , g g }
• { b b g , b g b , b g g , g b b , g b g , g g b , g g g }
• { b b b , b b g , b g b , g b b }
• { b g g , g b g , g g b }
• { g b b , g b g , g g b , g g g }

The relative frequencies for 1 through 6 are 0.16, 0.194, 0.162, 0.164, 0.154 and 0.166. It would appear that the die is not balanced.

Randomization in Statistics and Experimental Design

Design of Experiments > Randomization

What is Randomization?

Randomization in an experiment is where you choose your experimental participants randomly . For example, you might use simple random sampling , where participants names are drawn randomly from a pool where everyone has an even probability of being chosen. You can also assign treatments randomly to participants, by assigning random numbers from a random number table.

If you use randomization in your experiments, you guard against bias . For example, selection bias (where some groups are underrepresented) is eliminated and accidental bias (where chance imbalances happen) is minimized. You can also run a variety of statistical tests on your data (to test your hypotheses) if your sample is random.

Randomization Techniques

The word “random” has a very specific meaning in statistics. Arbitrarily choosing names from a list might seem random, but it actually isn’t. Hidden biases (like a subconscious preference for English names, names that sound like friends, or names that roll off the tongue) means that what you think is a random selection probably isn’t. Because these biases are often hidden, or overlooked, specific randomization techniques have been developed for researchers:

• 1. Probability Spaces

1. Random Experiments

Experiments.

Probability theory is based on the paradigm of a random experiment ; that is, an experiment whose outcome cannot be predicted with certainty, before the experiment is run. In classical or frequency-based probability theory, we also assume that the experiment can be repeated indefinitely under essentially the same conditions. The repetitions can be in time (as when we toss a single coin over and over again) or in space (as when we toss a bunch of similar coins all at once). The repeatability assumption is important because the classical theory is concerned with the long-term behavior as the experiment is replicated. By contrast, subjective or belief-based probability theory is concerned with measures of belief about what will happen when we run the experiment. In this view, repeatability is a less crucial assumption. In any event, a complete description of a random experiment requires a careful definition of precisely what information about the experiment is being recorded, that is, a careful definition of what constitutes an outcome .

The term parameter refers to a non-random quantity in a model that, once chosen, remains constant. Many probability models of random experiments have one or more parameters that can be adjusted to fit the physical experiment being modeled.

The subjects of probability and statistics have an inverse relationship of sorts. In probability, we start with a completely specified mathematical model of a random experiment. Our goal is perform various computations that help us understand the random experiment, help us predict what will happen when we run the experiment. In statistics, by contrast, we start with an incompletely specified mathematical model (one or more parameters may be unknown, for example). We run the experiment to collect data, and then use the data to draw inferences about the unknown factors in the mathematical model.

Compound Experiments

Suppose that we have \(n\) experiments \((E_1, E_2, \ldots, E_n)\). We can form a new, compound experiment by performing the \(n\) experiments in sequence, \(E_1\) first, and then \(E_2\) and so on, independently of one another. The term independent means, intuitively, that the outcome of one experiment has no influence over any of the other experiments. We will make the term mathematically precise in the section on independence .

In particular, suppose that we have a basic experiment. A fixed number (or even an infinite number) of independent replications of the basic experiment is a new, compound experiment. Many experiments turn out to be compound experiments and moreover, as noted above, (classical) probability theory itself is based on the idea of replicating an experiment.

In particular, suppose that we have a simple experiment with two outcomes. Independent replications of this experiment are referred to as Bernoulli trials , named for Jacob Bernoulli . This is one of the simplest, but most important models in probability. More generally, suppose that we have a simple experiment with \(k\) possible outcomes. Independent replications of this experiment are referred to as multinomial trials .

Sometimes an experiment occurs in well-defined stages, but in a dependent way, in the sense that the outcome of a given stage is influenced by the outcomes of the previous stages.

Sampling Experiments

In most statistical studies, we start with a population of objects of interest. The objects may be people, memory chips, or acres of corn, for example. Usually there are one or more numerical measurements of interest to us—the height and weight of a person, the lifetime of a memory chip, the amount of rain, amount of fertilizer, and yield of an acre of corn.

Although our interest is in the entire population of objects, this set is usually too large and too amorphous to study. Instead, we collect a random sample of objects from the population and record the measurements of interest of for each object in the sample.

There are two basic types of sampling. If we sample with replacement , each item is replaced in the population before the next draw; thus, a single object may occur several times in the sample. If we sample without replacement , objects are not replaced in the population. The chapter on finite sampling models explores a number of models based on sampling from a finite population.

Sampling with replacement can be thought of as a compound experiment, based on independent replications of the simple experiment of drawing a single object from the population and recording the measurements of interest. Conversely, a compound experiment that consists of \(n\) independent replications of a simple experiment can usually be thought of as a sampling experiment. On the other hand, sampling without replacement is an experiment that consists of dependent stages, because the population changes with each draw.

Examples and Applications

Probability theory is often illustrated using simple devices from games of chance: coins, dice, card, spinners, urns with balls, and so forth. Examples based on such devices are pedagogically valuable because of their simplicity and conceptual clarity. On the other hand, it would be a terrible shame if you were to think that probability is only about gambling and games of chance. Rather, try to see problems involving coins, dice, etc. as metaphors for more complex and realistic problems.

Coins and Dice

In terms of probability, the important fact about a coin is simply that when tossed it lands on one side or the other. Coins in Western societies, dating to antiquity, usually have the head of a prominent person engraved on one side and something of lesser importance on the other. In non-Western societies, coins often did not have a head on either side, but did have distinct engravings on the two sides, one typically more important than the other. Nonetheless, heads and tails are the ubiquitous terms used in probability theory to distinguish the front or obverse side of the coin from the back or reverse side of the coin.

Consider the coin experiment of tossing a coin \(n\) times and recording the score (1 for heads or 0 for tails) for each toss.

• Identify a parameter of the experiment.
• Interpret the experiment as a compound experiment.
• Interpret the experiment as a sampling experiment.
• Interpret the experiment as \(n\) Bernoulli trials.
• The number of coins \(n\) is the parameter.
• The experiment consists of \(n\) independent replications of the simple experiment of tossing the coin one time.
• The experiment can be thought of as selecting a sample of size \( n \) with replacement from he population \(\{0, 1\}\).
• There are two outcomes on each toss and the tosses are independent.

In the simulation of the coin experiment , set \(n = 5\). Run the simulation 100 times and observe the outcomes.

Dice are randomizing devices that, like coins, date to antiquity and come in a variety of sizes and shapes. Typically, the faces of a die have numbers or other symbols engraved on them. Again, the important fact is that when a die is thrown, a unique face is chosen (usually the upward face, but sometimes the downward one). We will study dice in more detail later.

Consider the dice experiment of throwing a \(k\)-sided die (with faces numbered 1 to \(k\)), \(n\) times and recording the scores for each throw.

• Identify the parameters of the experiment.
• Identify the experiment as \(n\) multinomial trials.
• The parameters are the number of dice \(n\) and the number of faces \(k\).
• The experiment consists of \(n\) independent replications of the simple experiment of throwing one die.
• The experiment can be thought of as selecting a sample of size \( n \) with replacement form the population \(\{1, 2, \ldots, k\}\).
• The same \( k \) outcomes occur for each die the throws are independent.

In reality, most dice are Platonic solids (named for Plato of course) with 4, 6, 8, 12, or 20 sides. The six-sided die is the standard die .

In the simulation of the dice experiment , set \(n = 5\). Run the simulation 100 times and observe the outcomes.

In the die-coin experiment , a standard die is thrown and then a coin is tossed the number of times shown on the die. The sequence of coin scores is recorded (1 for heads and 0 for tails). Interpret the experiment as a compound experiment.

The first stage consists rolling the die and the second stage consists of tossing the coin. The stages are dependent because the number of tosses depends on the outcome of the die throw.

Note that the experiment in can be obtained by randomizing the parameter \(n\) in the basic coin experiment in .

Run the simulation of the die-coin experiment 100 times and observe the outcomes.

In the coin-die experiment , a coin is tossed. If the coin lands heads, a red die is thrown and if the coin lands tails, a green die is thrown. The coin score (1 for heads and 0 for tails) and the die score are recorded. Interpret the experiment as a compound experiment.

The first stage consists of tossing the coin and the second stage consists of rolling the die. The stages are dependent because different dice (that may behave differently) are thrown, depending on the outcome of the coin toss.

Run the simulation of the coin-die experiment 100 times and observe the outcomes.

Playing cards, like coins and dice, date to antiquity. From the point of view of probability, the important fact is that a playing card encodes a number of properties or attributes on the front of the card that are hidden on the back of the card. In the next section thiese properties will become random variables . In particular, a standard card deck can be modeled by the Cartesian product set \[ D = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, j, q, k \} \times \{\clubsuit, \diamondsuit, \heartsuit, \spadesuit \} \] where the first coordinate encodes the denomination or kind (ace, 2–10, jack, queen, king) and where the second coordinate encodes the suit (clubs, diamonds, hearts, spades). Sometimes we represent a card as a string rather than an ordered pair (for example \(q \heartsuit\) rather than \((q, \heartsuit)\) for the queen of hearts). Some other properties, derived from the two main ones, are color (diamonds and hearts are red, clubs and spades are black), face (jacks, queens, and kings have faces, the other cards do not), and suit order (from least to highest rank: \( (\clubsuit, \diamondsuit, \heartsuit, \spadesuit) \)).

Consider the card experiment that consists of dealing \(n\) cards from a standard deck (without replacement).

• The parameter is \( n \), the number of cards dealt.
• At each stage, we draw a card from a deck, but the deck changes from one draw to the next, so the stages are dependent.
• The experiment is to select a sample of size \( n \) from the population \(D\), without replacement.

In the simulation of the card experiment , set \(n = 5\). Run the simulation 100 times and observe the outcomes.

The special case \(n = 5\) is the poker experiment and the special case \(n = 13\) is the bridge experiment .

Open each of the following to see depictions of card playing in some famous paintings.

• Cheat with the Ace of Clubs by Georges de La Tour
• The Cardsharps by Michelangelo Carravagio
• The Card Players by Paul Cézanne
• His Station and Four Aces by CM Coolidge
• Waterloo by CM Coolidge

Urn models are often used in probability as simple metaphors for sampling from a finite population.

An urn contains \(m\) distinct balls, labeled from 1 to \(m\). The experiment consists of selecting \(n\) balls from the urn, without replacement, and recording the sequence of ball numbers.

• The parameters are the number of balls \(m\) and the sample size \(n\).
• At each stage, we draw a ball from the urn, but the contents of the urn change from one draw to the next, so the stages are dependent
• The experiment is to select a sample of size \( n \) from the balls in the urn (the population), without replacement.

Consider the basic urn model of the previous exercise. Suppose that \(r\) of the \(m\) balls are red and the remaining \(m - r\) balls are green. Identify an additional parameter of the model. This experiment is a metaphor for sampling from a general dichotomous population

The parameters are the population size \(m\), the sample size \(n\), and the number of red balls \(r\).

In the simulation of the urn experiment , set \(m = 100\), \(r = 40\), and \(n = 25\). Run the experiment 100 times and observe the results.

An urn initially contains \(m\) balls; \(r\) are red and \(m - r\) are green. A ball is selected from the urn and removed, and then replaced with \(k\) balls of the same color. The process is repeated. This is known as Pólya's urn model , named after George Pólya .

• Interpret the case \(k = 0\) as a sampling experiment.
• Interpret the case \(k = 1\) as a sampling experiment.
• The parameters are the population size \(m\), the initial number of red balls \(r\), and the number new balls added \(k\).
• When \(k = 0\), each ball drawn is removed and no new balls are added, so the experiment is to select a sample of size \( n \) from the urn, without replacement.
• When \(k = 1\), each ball drawn is replaced with another ball of the same color. So at least in terms of the colors of the balls, the experiment is equivalent to selecting a sample of size \( n \) from the urn, with replacement.

Open the image of the painting Allegory of Fortune by Dosso Dossi. Presumably the young man has chosen lottery tickets from an urn.

Buffon's Coin Experiment

Buffon's coin experiment consists of tossing a coin with radius \(r \leq \frac{1}{2} \) on a floor covered with square tiles of side length 1. The coordinates of the center of the coin are recorded, relative to axes through the center of the square, parallel to the sides. The experiment is named for comte de Buffon .

• Identify a parameter of the experiment
• Interpret the experiment as sampling experiment.
• The parameter is the coin radius \(r\).
• The experiment can be thought of as selecting the coordinates of the coin center independently of one another.
• The experiment is equivalent to selecting a sample of size 2 from the population \(\left[-\frac{1}{2}, \frac{1}{2}\right]\), with replacement.

In the simulation of Buffon's coin experiment , set \(r = 0.1\). Run the experiment 100 times and observe the outcomes.

Reliability

In the usual model of structural reliability , a system consists of \(n\) components, each of which is either working or failed . The states of the components are uncertain, and hence define a random experiment. The system as a whole is also either working or failed, depending on the states of the components and how the components are connected. For example, a series system works if and only if each component works, while a parallel system works if and only if at least one component works. More generally, a \(k\) out of \(n\) system works if at least \(k\) components work.

Consider the \(k\) out of \(n\) reliability model.

• Identify two parameters.
• What value of \(k\) gives a series system?
• What value of \(k\) gives a parallel system?
• The parameters are \(k\) and \(n\)
• \(k = n\) gives a series system.
• \(k = 1\) gives a parallel system.

The reliability model above is a static model. It can be extended to a dynamic model by assuming that each component is initially working, but has a random time until failure. The system as a whole would also have a random time until failure that would depend on the component failure times and the structure of the system.

In ordinary sexual reproduction, the genetic material of a child is a random combination of the genetic material of the parents. Thus, the birth of a child is a random experiment with respect to outcomes such as eye color, hair type, and many other physical traits. We are often particularly interested in the random transmission of traits and the random transmission of genetic disorders.

For example, let's consider an overly simplified model of an inherited trait that has two possible states ( phenotypes ), say a pea plant whose pods are either green or yellow. The term allele refers to alternate forms of a particular gene, so we are assuming that there is a gene that determines pod color, with two alleles: \(g\) for green and \(y\) for yellow. A pea plant has two alleles for the trait (one from each parent), so the possible genotypes are

• \(gg\), alleles for green pods from each parent.
• \(gy\), an allele for green pods from one parent and an allele for yellow pods from the other (we usually cannot observe which parent contributed which allele).
• \(yy\), alleles for yellow pods from each parent.

The genotypes \(gg\) and \(yy\) are called homozygous because the two alleles are the same, while the genotype \(gy\) is called heterozygous because the two alleles are different. Typically, one of the alleles of the inherited trait is dominant and the other recessive . Thus, for example, if \(g\) is the dominant allele for pod color, then a plant with genotype \(gg\) or \(gy\) has green pods, while a plant with genotype \(yy\) has yellow pods. Genes are passed from parent to child in a random manner, so each new plant is a random experiment with respect to pod color.

Pod color in peas was actually one of the first examples of an inherited trait studied by Gregor Mendel , who is considered the father of modern genetics. Mendel also studied the color of the flowers (yellow or purple), the length of the stems (short or long), and the texture of the seeds (round or wrinkled).

For another example, the \(ABO\) blood type in humans is controlled by three alleles: \(a\), \(b\), and \(o\). Thus, the possible genotypes are \(aa\), \(ab\), \(ao\), \(bb\), \(bo\) and \(oo\). The alleles \(a\) and \(b\) are co-dominant and \(o\) is recessive . Thus there are four possible blood types (phenotypes):

• Type \(A\): genotype \(aa\) or \(ao\)
• Type \(B\): genotype \(bb\) or \(bo\)
• Type \(AB\): genotype \(ab\)
• type \(O\): genotype \(oo\)

Of course, blood may be typed in much more extensive ways than the simple \(ABO\) typing. The RH factor (positive or negative) is the most well-known example.

For our third example, consider a sex-linked hereditary disorder in humans. This is a disorder due to a defect on the X chromosome (one of the two chromosomes that determine gender). Suppose that \(h\) denotes the healthy allele and \(d\) the defective allele for the gene linked to the disorder. Women have two X chromosomes, and typically \(d\) is recessive . Thus, a woman with genotype \(hh\) is completely normal with respect to the condition; a woman with genotype \(hd\) does not have the disorder, but is a carrier , since she can pass the defective allele to her children; and a woman with genotype \(dd\) has the disorder. A man has only one X chromosome (his other sex chromosome, the Y chromosome , typically plays no role in the disorder). A man with genotype \(h\) is normal and a man with genotype \(d\) has the disorder. Examples of sex-linked hereditary disorders are dichromatism , the most common form of color-blindness, and hemophilia , a bleeding disorder. Again, genes are passed from parent to child in a random manner, so the birth of a child is a random experiment in terms of the disorder.

Point Processes

There are a number of important processes that generate random points in time . Often the random points are referred to as arrivals . Here are some specific examples:

• times that a piece of radioactive material emits elementary particles
• times that customers arrive at a store
• times that requests arrive at a web server
• failure times of a device

To formalize an experiment, we might record the number of arrivals during a specified interval of time or we might record the times of successive arrivals.

There are other processes that produce random points in space . For example,

• flaws in a piece of sheet metal
• errors in a string of symbols (in a computer program, for example)
• raisins in a cake
• misprints on a page
• stars in a region of space

Again, to formalize an experiment, we might record the number of points in a given region of space.

Statistical Experiments

In 1879, Albert Michelson constructed an experiment for measuring the speed of light with an interferometer. The velocity of light data set contains the results of 100 repetitions of Michelson's experiment. Explore the data set and explain, in a general way, the variability of the data.

The variablility is due to measurement and other experimental errors beyond the control of Michelson.

In 1998, two students at the University of Alabama in Huntsville designed the following experiment: purchase a bag of M&Ms (of a specified advertised size) and record the counts for red, green, blue, orange, and yellow candies, and the net weight (in grams). Explore the M&M data. set and explain, in a general way, the variability of the data.

The variability in weight is due to measurement error on the part of the students and to manufacturing errors on the part of the company. The variability in color counts is less clear and may be due to purposeful randomness on the part of the company.

In 1999, two researchers at Belmont University designed the following experiment: capture a cicada in the Middle Tennessee area, and record the body weight (in grams), the wing length, wing width, and body length (in millimeters), the gender, and the species type. The cicada data set contains the results of 104 repetitions of this experiment. Explore the cicada data and explain, in a general way, the variability of the data.

The variability in body measurements is due to differences in the three species, to all sorts of envirnomental factors, and to measurement errors by the researchers.

On June 6, 1761, James Short made 53 measurements of the parallax of the sun, based on the transit of Venus. Explore the Short data set and explain, in a general way, the variability of the data.

The variability is due to measurement and other experimental errors beyond the control of Short.

In 1954, two massive field trials were conducted in an attempt to determine the effectiveness of the new vaccine developed by Jonas Salk for the prevention of polio. In both trials, a treatment group of children were given the vaccine while a control group of children were not. The incidence of polio in each group was measured. Explore the polio field trial data set and explain, in a general way, the underlying random experiment.

The basic random experiment is to observe whether a given child, in the treatment group or control group, comes down with polio in a specified period of time. Presumabley, a lower incidence of polio in the treatment group compared with the control group would be evidence that the vaccine was effective.

Each year from 1969 to 1972 a lottery was held in the US to determine who would be drafted for military service. Essentially, the lottery was a ball and urn model and became famous because many believed that the process was not sufficiently random. Explore the Vietnam draft lottery data set and speculate on how one might judge the degree of randomness.

This is a difficult problem, but presumably in a sufficiently random lottery, one would not expect to see dates in the same month clustered too closely together. Observing such clustering, then, would be evidence that the lottery was not random.

Deterministic Versus Probabilistic Models

One could argue that some of the examples discussed above are inherently deterministic . In tossing a coin, for example, if we know the initial conditions (involving position, velocity, rotation, etc.), the forces acting on the coin (gravity, air resistance, etc.), and the makeup of the coin (shape, mass density, center of mass, etc.), then the laws of physics should allow us to predict precisely how the coin will land. This is true in a technical, theoretical sense, but false in a very real sense. Coins, dice, and many more complicated and important systems are chaotic in the sense that the outcomes of interest depend in a very sensitive way on the initial conditions and other parameters. In such situations, it might well be impossible to ever know the initial conditions and forces accurately enough to use deterministic methods.

In the coin experiment, for example, even if we strip away most of the real world complexity, we are still left with an essentially random experiment. Joseph Keller in his article The Probability of Heads deterministically analyzed the toss of a coin under a number of ideal assumptions:

• The coin is a perfect circle and has negligible thickness
• The center of gravity of the coin is the geometric center.
• The coin is initially heads up and is given an initial upward velocity \(u\) and angular velocity \(\omega\).
• In flight, the coin rotates about a horizontal axis along a diameter of the coin.
• In flight, the coin is governed only by the force of gravity. All other possible forces (air resistance or wind, for example) are neglected.
• The coin does not bounce or roll after landing (as might be the case if it lands in sand or mud).

Of course, few of these ideal assumptions are valid for real coins tossed by humans. Let \(t = u / g\) where \(g\) is the acceleration of gravity (in appropriate units). Note that the \(t\) just has units of time (in seconds) and hence is independent of how distance is measured. The scaled parameter \(t\) actually represents the time required for the coin to reach its maximum height.

Keller showed that the regions of the parameter space \((t, \omega)\) where the coin lands either heads up or tails up are separated by the curves \[ \omega = \left( 2n \pm \frac{1}{2} \right) \frac{\pi}{2t}, \quad n \in \N \] The parameter \(n\) is the total number of revolutions in the toss. A plot of some of these curves is given below. The largest region, in the lower left corner, corresponds to the event that the coin does not complete even one rotation, and so of course lands heads up, just as it started. The next region corresponds to one rotation, with the coin landing tails up. In general, the regions alternate between heads and tails.

The important point, of course, is that for even moderate values of \(t\) and \(\omega\), the curves are very close together, so that a small change in the initial conditions can easily shift the outcome from heads up to tails up or conversely. As noted in Keller's article, the probabilist and statistician Persi Diaconis determined experimentally that typical values of the initial conditions for a real coin toss are \(t = \frac{1}{4}\) seconds and \(\omega = 76 \pi \approx 238.6\) radians per second. These values correspond to \(n = 19\) revolutions in the toss. Of course, this parameter point is far beyond the region shown in our graph, in a region where the curves are exquisitely close together.

• Random Experiment

Let’s say that you toss a fair coin (not a prank coin ). There are only two possible outcomes – a head or a tail. Also, it is impossible to accurately predict the outcome (a head or a tail). In mathematical theory, we consider only those experiments or observations, for which we know the set of possible outcomes. Also, it is important that predicting a particular outcome is impossible. Such an experiment, where we know the set of all possible results but find it impossible to predict one at any particular execution, is a random experiment.

Suggested Videos

Even if a random experiment is repeated under identical conditions, the outcomes or results may fluctuate or vary randomly. Let’s look at another example – you take a fair dice and roll it using a box.

When the dice lands there are only six possible outcomes – 1, 2, 3, 4, 5, or 6. However, predicting which one will occur at any roll of the dice is completely unpredictable.

                                                                                                                                                 Source: Maxpixel

Further, in any experiment, there are certain terms that you need to know:

• Trial – A trial is the performance of an experiment.
• Outcomes – Whenever you perform an experiment, you get an outcome. For example, when you flip a coin, the outcome is either heads or tails. Similarly, when you roll a dice, the outcome is 1, 2, 3, 4, 5, or 6.
• Event – An event is a collection of basic outcomes with specific properties. For example, ‘E’ is the event where our roll of a six-sided dice has an outcome of less than or equal to 3. Therefore, E is the collection of basic outcomes where the result is 3 or less. Symbolically, E = {O 1 , O 2 , O 3 }. It is important to note that depending on the event , the outcomes can be of any number (even zero).

Random Experiment – Types of Events

In a random experiment, the following types of events are possible:

Simple and Compound Events

Simple or Elementary events are those which we cannot decompose further. For example, when you toss a coin, there are only two possible outcomes (heads or tails).

The event that the toss turns up a ‘head’ is a simple event and so is the event of it turning up a ‘tail’. Similarly, when you roll a six-sided dice, then the event that number 3 comes up is a simple event.

The Compound or Composite events are those which we can decompose into elementary or simple events. In simpler words , an elementary event corresponds to a single possible outcome of an experiment.

On the other hand, a compound event is an aggregate of some elementary events and we can decompose it into simple events.

To give you some examples, when you toss a fair coin, the event ‘turning up of a head or a tail’ is a compound event. This is because we can decompose this event into two simple events – (i) turning up of the head and (ii) turning up of the tail.

Similarly, when you roll a six-sided dice, the event that an odd number comes up is a compound event. This is because we can break it down into three simple events – (i) Number 1 comes up, (ii) Number 3 comes up, and (iii) The third odd number 5 comes up.

Learn more about Random Variables here in detail.

Equally Likely Events

If among all possible events, you cannot expect either one to occur in preference in the same experiment, after taking all conditions into account, then the events are Equally Likely Events.

Examples of Equally Likely Events

Back to our favorite coin. Tossing a fair coin has two simple events associated with it. The coin will turn up a ‘head’ or a ‘tail’. Now, there is an equal chance of either turning up and you cannot expect one to turn up more frequently than the other. Also, in the case of rolling a six-sided dice, there are six equally likely events.

Mutually Exclusive Events

In a random experiment, if the occurrence of one event prevents the occurrence of any other event at the same time, then these events are Mutually Exclusive Events.

Examples of Mutually Exclusive Events

Let’s call the coin back into action, shall we?

When you toss a fair coin, the turning up of heads and turning up of tails are two mutually exclusive events. This is because if one turns up, then the other cannot turn up in the same experiment.

Similarly, when you roll a six-sided dice, there are six mutually exclusive events .

Remember, mutually exclusive events cannot occur simultaneously in the same experiment. Also, they may or may not be equally likely.

Independent Events

Two or more events are Independent Events if the outcome of one does not affect the outcome of the other. For example, (coin again!) if you toss a coin twice, then the result of the second throw is not affected by the result of the first throw.

Dependent Events

Two or more events are Dependent Events if the occurrence or non-occurrence of one in any trial affects the probability of the other events in other trials.

Examples of Dependent Events

No coin this time.

Let’s say that the event is drawing a Queen from a pack of 52 cards. When you start with a new deck of cards, the probability of drawing a Queen is 4/52. However, if you manage to draw a Queen in one trial and do not replace the card in the pack, then the probability of drawing a Queen in the remaining trials becomes 3/51.

Exhaustive Events

In order to understand Exhaustive Events, let’s take a quick look at the concept of Sample Space.

Sample Space

The Sample Space (S) of an experiment is the set of all possible outcomes of the experiment.

Going back to our coin, the sample space is S = {H, T} … where H-heads and T-tails. Similarly, when you roll a six-sided dice, the sample space is S = {1, 2, 3, 4, 5, 6}. Every possibility is a sample point or element of the sample space.

Further, an event is a subset of the sample space and can contain one or more sample points. For example, when you roll a dice, the event that an odd number appears has three sample points.

Coming back to exhaustive events, the total number of possible outcomes of a random experiment form an exhaustive set of events. In other words, events are exhaustive if we consider all possible outcomes.

Solved Question

Q1. What is a Random Experiment?

Answer: An experiment, where we know the set of all possible results but find it impossible to predict one at any particular execution, is a random experiment.

Customize your course in 30 seconds

Which class are you in.

Mathematics of Finance and Elementary Probability

• Expected Values
• Approaches of Probability
• Random Variables
• Present Value

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

Download the App