Introduction to Probability: Random experiment, Sample space, Types of Events, Practice problems and FAQs

We all have witnessed that a coin is tossed before starting a cricket match or a football match.

Can you tell whether it will be a head or a tail?

Yes, you can predict the possibility of the outcome but can’t be sure. In mathematical terms, possibility is known as probability. There are many real life situations in which we have to predict the outcome of an event like in games and business. When we are not sure about the results of an event, in such cases, we say that there is a probability of this event to occur. Let us learn about probability in detail in this article.

Table of contents

• Introduction
• Random experiment
• Sample space
• Types of Events
• Algebra of events
• Practice Problems
• FAQs

Introduction

Probability can be defined as the ratio of number of favourable outcomes to the number of total outcomes. For an event E the probability is denoted by P(E) which is read as probability of E. Probability of event E can be mathematically described as:

P(E)=Number of outcomes favourable to event ETotal number of outcomes

Let us learn about some basic terminology of probability in this article.

Experiment

In Probability theory an operation which can produce well defined outcomes is called an experiment.

Random experiment

A random experiment is defined as an experiment in which, for each trail the outcome may be any one of the possible outcomes, when performed under identical conditions.

Consider an experiment of identifying number of people of height 12 feet.It’s not random experiment as the outcome is always zero which is predictable.

A Random experiment should satisfy the following two conditions:

• It must have more than one possible outcome
• Prediction of the exact outcome in advance is not possible

Following are some of the examples of random experiments:

1. Tossing a coin
2. Throwing a pair of dice
3. Drawing a card from a deck of 52 cards

Note: When an experiment is performed once, it is called a trial.

Sample space

A sample space is the set of all possible outcomes of a random experiment. It is generally denote by S. Each outcome is called an elementary event.

Example

1. When a coin is tossed, either a head or a tail will come up. If H denotes the occurrence of head and T denotes the occurrence of tail, then the sample space for tossing a coin, S={H, T}.
2. When two coins are tossed once, denoting head and tail by H and T respectively, the sample space is S={HH, HT, TH, TT}.
3. When a dice and a coin are thrown simultaneously, then the sample space is S={(1,H),(2,H),(3,H),(4,H),(5,H),(6,H),(1,T),(2,T),(3,T),(4,T),(5,T),(6,T)}

Note:

• When a random experiment has m outcomes and another one has n outcomes then the sample space of the two experiments together will have mn outcomes.
• Each element of a sample space is called a sample point or an elementary event.

Event

Each subset of a sample space is known as an event or a case.

In other words, a sub-collection of a number of sample points under a fixed/defined rule is called an event.

Example

When a dice is thrown, the sample space is S={1,2,3,4,5,6}. Let an even number is to be obtained. Denoting this event by E, we have E={2,4,6}.

Different types of events:

Null Event

An event having no sample point is called null event. It is denoted by . It is also known as impossible event. For example, when a dice is thrown, the event E= getting a number greater than 6, is a null event.

Simple or Elementary Event

If an event A has only one sample point of a sample space, it is called an elementary(or simple) event. For example, when a coin is tossed two times, sample space is S={HH,HT,TH,TT}. Let A={TT}= the event of occurrence of two tails and

B={TH}= the event of occurrence in which results of first and second tosses are T and H, respectively.

Here, A and B are simple events.

Mixed or Compound or Composite Event

A subset of the sample space S which contains more than one element is called a composite or mixed event. For example, in the experiment of tossing a coin thrice, the event

E : atleast one head appeared, given by E={HTT,THT,TTH,HHT,HTH,THH,HHH} is a compound event.

Equally Likely Events

Cases(or events) are said to be equally likely when we do not expect the happening of one event in preference to the other. For example, in case of throwing a fair dice all the faces, 1, 2, 3, 4, 5 and 6 are equally likely to come up.

Exhaustive Events(Cases)

If E1,E2,E3,...,En are n events of a sample space S and if E1E2E3...En=i=1nEi=S, then E1,E2,E3,...,En are called exhaustive events. In simpler words we can say that, if on combining n events, we get the sample space, then those events will be called exhaustive events.

For example,

Consider the experiment of throwing a dice. We have S={1,2,3,4,5,6}. let us define the following events:

A : a number less than 3 appears

B : a number greater than 1 but less than 5 appears

C : a number greater than 2 appears

Then, A={1,2}, B={2,3,4} and C={3,4,5,6}.

We observe that ABC={1,2}{2,3,4}{3,4,5,6}={1,2,3,4,5,6}=S

Here events A,B and C are called exhaustive events.

Mutually Exclusive Events

Two or more events are said to be mutually exclusive if nothing is common in them. Two events E and F are called mutually exclusive events if the occurrence of any one of them excludes the occurrence of the other, i.e., they cannot occur simultaneously.

For example, in the experiment of rolling a dice, the sample space is S={1,2,3,4,5,6}. Consider events, E= ’an odd number appears’ and F= ’an even number appears’. Clearly, the occurrence of event E excludes the occurrence of F and vice versa. Hence E and F are mutually exclusive events.

For mutually exclusive events A,B and C, AB=BC=AC=ABC=.

Note: Two events which are not mutually exclusive are known as compatible events.

Algebra of Events

In set theory, we have learnt how to perform operations such as, union, intersection, difference, symmetric difference etc. on two or more sets. Similarly, we can combine two or more events by using the analogous set notations. Let’s understand some of these here.

Let A and B be two events associated with random experiment whose sample space is S.

1. Event ‘A or B’: We know that union of two sets A and B(denoted by AB) contains all those elements which are either in A or in B or in both. Likewise, the event ‘A or B’ is also represented as AB or A+B.

For example, let a dice be thrown and A denotes the event of getting an odd number whereas B denotes the event of getting a number greater than 3. Then, A={1,3,5}, B={4,5,6}. Thus, A or B=AB={1,3,4,5,6}.

2. Event ‘A and B’: We know that intersection of two sets A and B is the set of those elements which are common to both A and B, i.e., belong to both A and B. Likewise, the event ‘ A and B’ is represented by AB.

For example, in the figure below, occurrence of 9 is common in both the events. Hence, AB={9}.

3. Event ‘A but not B’: We know that A-B is the set of all those elements which are in A but not in B. Therefore, the set A-B denotes the event ‘A but not B’. Also, A-B is same as AB' or A-(AB). For example, in the example used above A={1,6,9}, B={2,4,9}, thus A-B={1,6}.

4. Complementary Event or the event ‘not A’

For every event A, there exists another event A' called the complementary event to A. It is also called the event ‘not A’. For example, take the experiment of tossing three coins. The sample space is S={HHH,HHT,HTH,THH,HTT,THT,TTH,TTT}.

Let A={HTH,HHT,THH} be the event ‘tail appears only once’

then , the complementary event ‘not A’ is A'={HHH,HTT,THT,TTH,TTT}.

Practice Problems

1.  A bag contains 3 identical red balls and 4 identical black balls. The experiment consists of drawing one ball, then putting it into the bag and again drawing a ball. What are the possible outcomes of the experiment?

Solution:  Let’s denote the red ball by R and black ball by B

Then, the sample space S of drawing one ball, then putting it into the bag and again drawing a ball is given by S={RR,RB,BR,BB}.

2. A coin is tossed twice. If the second throw results in a head, a die is thrown, otherwise a coin is tossed again. Write the sample space.

Solution: When a coin is tossed twice, the possible outcomes are {HH,HT,TH,TT}.

Now, it is given that when the second throw results in a head(i.e., HH,TH) a dice is thrown. Then the possible outcomes are

{HH1,HH2,HH3,HH4,HH5,HH6,TH1,TH2,TH3,TH4,TH5,TH6}

Otherwise, when the second throw does not results in a head(i.e., HT,TT), a coin is tossed. Then the possible outcomes are: {HTH,HTT,TTH,TTT}.

Thus, the required sample space is {HH1,HH2,HH3,HH4,HH5,HH6,TH1,TH2,TH3,TH4,TH5,TH6,HTH,HTT,TTH,TTT}.

3. A die is thrown once. Let E be the event of getting a prime number, F be the event of getting an odd number. Write the sets representing the following events:

(i) E or F (ii) E and F (iii) E but not F (iv) not E

Solution: When a die is thrown, then the number of possible outcomes is 6, so

S={1,2,3,4,5,6}

E : getting a prime number ={2,3,5}

F : getting an odd number ={1,3,5}

Then,

(i) E or F=EF={2,3,5}{1,3,5}={1,2,3,5}

(ii) E and F=EF={2,3,5}{1,3,5}={3,5}

(iii) E but not F=E-F={2,3,5}-{1,3,5}={2}

(iv) not E=E'=S-E={1,2,3,4,5,6}-{2,3,5}={1,4,6}

4. A coin is tossed three times. Consider the following events:

X : no head appears

Y : exactly one head appears

Z : atleast two head appears

Do events X,Y,Z form a set of mutually exclusive and exhaustive events?

Solution:  When a coin is tossed three times, the sample space has 8 elements given by

S={HHH,HHT,HTH,THH,HTT,THT,TTH,TTT}

X : no head appears ={TTT}

Y : exactly one head appears ={HTT,THT,TTH}

Z : atleast two head appears ={HHT,HTH,THH,HHH}

Therefore, XYZ={HHH,HHT,HTH,THH,HTT,THT,TTH,TTT}=S

Thus, X,Y and Z are exhaustive events.

Also, XY=ZY=XZ=XYZ=

Therefore, the events are mutually exclusive.

Hence, X,Y,Z form a set of mutually exclusive and exhaustive events.

FAQs

1. What is the complement of null event?
Answer: Null event is denoted by . complement of null event ='=S-=S. Hence complement of null event will be the sample space itself

2. What is the probability of occurrence of sample space?
Answer: A sample space consists of all the possible outcomes of an experiment, so its probability is always 1.

3. What are the different types of probability?
Answer: Different types of probability are theoretical probability, experimental probability and axiomatic probability.

4. Let n events associated to a random experiment are mutually exclusive. Is it necessary that every pair of these events should be mutually exclusive?
Answer: Yes, every pair of these n events should be mutually exclusive.