Types of Events in Probability | Algebra of Events in Probability | Solved Examples, FAQs

Class 11 Mathematics | Written by Neeraj Anand
Published by ANAND TECHNICAL PUBLISHERS

Probability is a branch of mathematics that deals with predicting the likelihood of various outcomes. In Class 11, understanding the types of events is crucial for solving problems in board exams and competitive exams like JEE Mains and Advanced.

In probability, events refer to the outcomes or sets of outcomes from an experiment.

What are Events?

A probability event can be defined as a set of outcomes of an experiment. For example, getting a head in a coin toss is an event and all the odd-numbered outcomes while rolling a die also constitute an event. 

An event is a subset of the sample space.

Sample Space : The entire possible set of outcomes of a random experiment is the sample space or the individual space of that experiment. For Example, the sample space for the tossing of three coins simultaneously is given by:

S = {(T , T , T) , (T , T , H) , (T , H , T) , (T , H , H ) , (H , T , T ) , (H , T , H) , (H , H, T) ,(H , H , H)}

Suppose, if we want to find only the outcomes which have at least two heads; then the set of all such possibilities can be given as:

E = { (H , T , H) , (H , H ,T) , (H , H ,H) , (T , H , H)}

Thus, an event is a subset of the sample space, i.e., E is a subset of S.

What is the Probability of Occurrence of an Event?

The likelihood of occurrence of an event is known as probability. The number of favourable outcomes to the total number of outcomes is defined as the probability of occurrence of any event. So, the probability that an event will occur is given as:

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

The probability of occurrence of any event lies between 0 and 1.

Type of Events

The following list gives the different types of events: 

Impossible Event 

Consider an experiment in which we roll a die. Now let’s define an event that consists of outcomes that are multiple of 11. Sample space for this event is denoted by S, 

S = {1, 2, 3, 4, 5, 6} 

Now since there is no outcome in the sample space which is a multiple of 11. So, the set of event E will be an empty set. 

The probability of occurrence of an impossible event is 0.

The impossible events are described by an empty set ∅ and the probability of occurrence of an impossible event is 0.

Sure Event

The probability of occurrence of a sure event is 1 i.e., occurrence of the event is certain or universal truth then that event is called Sure Event or Certain Event.

For example, If we roll a die, as the event is the occurrence of a number less than 7, then it is sure that the occurring number is always less than 7 as the die only has numbers 1, 2, 3, 4, 5, and 6. 

Note: The collection of elements from sample space in Sure Event is the complete Sample Space.

Simple Event

Any event that comprises a single result from the sample space is known as a simple event. 

For example, the Sample space of rolling a die, S= {1, 2, 3, 4, 5, 6} and the event for getting less than 2, E= {1}, where E has a single result taken from the sample space, Hence the event is a Simple event.

Compound Event

Contrary to the simple event, that is, any event that comprises more than a single result or more than a single point from the sample space, that event is known as a Compound event. 

For example, the Sample space of rolling a die, S= {1, 2, 3, 4, 5, 6} and the event for getting less than 4, E= {1, 2, 3}, where E is a Compound event.

Dependent Events

Dependent events are those in which the next outcome depends on the previous outcomes, which means, the probability of an event will change based on its previous outcomes.

For example, let’s take the example of drawing balls from a bag, there are 4 black and 3 red balls in a bag, a ball is drawn, and it came out to be black (In the first draw, the probability of a black ball was 4/7= 0.571. When a ball is drawn the next time, the probability of the black ball occurring will change as now there are fewer balls in the bag (3 black and 3 red balls are left), hence, now the probability of getting a black ball will be 3/6= 0.5. Thus, this event is dependent as the probability of each successive event depends upon the previous event.

Note : In the example above, there is a way of converting this dependent event into independent event, it can be done through Replacement. If after each experiment the ball is again kept in the bag, the sample space of the experiment will not change and hence, the probability of the event will remain same too.

Independent Event

Independent events are those in which the next outcome is independent of the previous outcome. This means the probability of the occurrence of an event will remain the same no matter how many times the same experiment is done. 

For example, let’s take the example of rolling a die, a die is rolled once and the probability of getting an odd number is 3/6 = 0.5, now the dice is rolled again, still the probability of getting an odd number will be 3/6 = 0.5 only. This means, that the probability of the event is independent of its previous outcomes.

Equally Likely Events

Those outcomes of an experiment that have the same probability are called Equally Likely Events. In other words, if two or more events have the same likelihood of happening, they are considered equally likely events.

For example, consider rolling a die. Each of the six possible outcomes (1, 2, 3, 4, 5, and 6) has the same probability of occurring, which is 1/6. Therefore, rolling a 1 is equally likely as rolling a 2, 3, 4, 5, or 6.

Similarly, consider drawing a card from a standard deck of 52 cards. There are 13 cards of each suit (hearts, diamonds, clubs, and spades) and 4 cards of each rank (ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and king). Therefore, the probability of drawing any particular card is 1/52, and the probability of drawing any particular suit or rank is 1/4.

Algebra of Events

Two or more sets can be combined using four different operations, union, intersection, difference, and compliment. Let’s consider three events A, B, and C defined over the sample space S. 

Complimentary Event 

For any event A there exists another event A‘ which represents the remaining elements of the sample space S i.e.

A = S − A‘

The event A’ is called a complimentary event.  It consists of all those elements which do not belong to event A.

For example, If a dice is rolled then the sample space S is given as S = {1 , 2 , 3 , 4 , 5 , 6 }. If event A represents all the outcomes which is greater than 4, then A = {5, 6} and the complementary A’ of event A will be consists of all the elements in the sample space which are not in event A. Thus, A‘ = {1, 2, 3, 4}.

Thus A‘ is the complement of the event A.

Similarly, the complement of E₁, E₂, E₃……….E_n will be represented as E₁‘, E₂‘, E₃‘……….E_n‘

Event A OR B

If two events A and B are associated with OR then it means set contains all the elements which are in either set A, set B, or both. This operation is called union operation. The Union of two sets A and B is denoted as A ∪ B.This event A or B is defined as, 

Event A or B = A ∪ B

OR

A ∪ B = {w : w ∈ A or w ∈ B}

If we have mutually exhaustive events E₁, E₂, E₃ ………E_n associated with sample space S then,

E₁ U E₂ U E₃ U ………E_n = S

Events A AND B

If two events A and B are associated with AND then it means the intersection of elements which is common to both the events. The intersection of two sets A and B is denoted as A ∩ B. This contains all the elements which are in both set A and set B. This event A and B is defined as, 

Event A and B = A ∩ B

OR

A ∩ B= {w: w ∈ A and w ∈ B}

Event A but not B

It represents the difference between both the events. The set difference A – B consists of all the elements which are in A but not in B. The events A but not B are defined as, 

A but not B = A – B 

OR

A – B = A ∩ B’

Where B’ is the complement of event B.

Using these concepts two other types of events are defined, that are:

• Mutually Exclusive Events
• Exhaustive Events

Let’s understand these two events as follows:

Mutually Exclusive Events

If the two events have nothing in common, then they are called mutually exclusive events. Thus two events A and B are called mutually exclusive if both of them cannot occur simultaneously. In this case, sets A and B are disjoint.

A ∩ B = ∅ 

For example, consider rolling a die, 

S = {1, 2, 3, 4, 5, 6} 

Now, event A is defined as “getting an even number” while event B is defined as “getting an odd number”. Now, these two events cannot occur together. 

A = {2, 4, 6} and B = {1, 3, 5}. 

Thus, the intersection between these two sets is an empty set. 

Exhaustive Events

Exhaustive events are those mutually exclusive events that together cover all the possible outcomes of an experiment. Let’s consider three events A, B, and C will be called exhaustive events if, 

A ∪ B ∪ C = S

In a more general setting, n events such that E₁, E₂,. . ., E_n is called exhaustive events if, 

 E₁ ∪ E₂ ∪. . .∪ E_n = S 

As an example, let’s say for a two-times coin toss experiment, 

A = Getting at least One head. 

B = Getting two tails. 

A = {HT, TH, HH} and B = {TT} 

Thus, A ∪ B = S

Example Question on Probability of Events

Case Study

In the game of snakes and ladders, a fair die is thrown. If event E₁ represents all the events of getting a natural number less than 4, event E₂ consists of all the events of getting an even number and E₃ denotes all the events of getting an odd number. List the sets representing the following:

i)E₁ or E₂ or E₃

ii)E₁ and E₂ and E₃

iii)E₁ but not E₃

Solution:

The sample space is given as S = {1 , 2 , 3 , 4 , 5 , 6}

E₁ = {1,2,3}

E₂ = {2,4,6}

E₃ = {1,3,5}

i)E₁ or E₂ or E₃= E₁ E₂ E₃= {1, 2, 3, 4, 5, 6}

ii)E₁ and E₂ and E₃ = E₁ E₂ E₃ = ∅

iii)E₁ but not E₃ = {2}

Sample Problems on Types of Events

Problems 1: Consider the experiment of tossing a fair coin 3 times, Event A is defined as getting all tails. What kind of event is this? 

Solution: 

Sample space for the coin toss will be, 

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

For the event A, 

A = {TTT}

This event is only mapped to one element of sample space. Thus, it is a simple event. 

Problems 2: Let’s say a coin is tossed once, state whether the following statement is True or False. 

“If we define an event X which means getting both heads and tails. This event will be a simple event.”

Solution:

When a coin it tossed, there can be only two outcomes, Heads or Tails. 

S = {H, T} 

Getting both Heads and Tails is not possible, thus event X is an empty set. 

Thus, it is an impossible and sure event. So, this statement is False. 

Problems 3: A die is rolled, and three events A, B, and C are defined below:

1. A: Getting a number greater than 3 
2. B: Getting a number that is multiple of 3. 
3. C: Getting an odd number

Find A ∩ B, A ∩ B ∩ C, and A ∪ B.

Solution:

Sample space for die roll will be, 

S = {1, 2, 3, 4, 5, 6} 

For the event A, 

A = {4, 5, 6}

For the event B, 

B = {3, 6}

For the event C, 

C = {1, 3, 5}

A ∩ B = {4, 5, 6} ∩ {3, 6}

⇒ A ∩ B = {6}

A ∩ B ∩ C = {4, 5, 6} ∩ {3, 6} ∩ {1, 3, 5}

⇒ A ∩ B ∩ C = ∅ (Empty Set) 

A ∪ B = {4, 5, 6} ∪ {3, 6}

⇒ A ∪ B = {3, 4, 5, 6}

Problems 4: A die is rolled, let’s define two events, event A is getting the number 2 and Event B is getting an even number. Are these events mutually exclusive? 

Solution: 

Sample space for die roll will be, 

S = {1, 2, 3, 4, 5, 6} 

For the event A, 

A = {2}

For the event B, 

B = {2, 4, 6}

For two events to be mutually exclusive, their intersection must be an empty set 

A ∩ B = {2} ∩ {2, 4, 6}

⇒ A ∩ B  = {2}

Since it is not an empty set, these events are not mutually exclusive.

FAQs on Types of Events in Probability

What is an Event in Probability?

An event in probability is a set of outcomes of a random experiment or in other words, an event in probability is the subset of the sample space.

What are the Different Types of Events in Probability?

The different types of events in probability are as follows:

• Impossible and Sure Events
• Simple Event and Compound Event
• Dependent and Independent Events
• Mutually Exclusive Events
• Exhaustive Events
• Equally Likely Events

What is a Simple Event?

A simple event is an event that consists of a single outcome. For example, if a coin is tossed, the event of getting a head is a simple event.

What is a Compound Event?

A compound event is an event that consists of two or more outcomes. For example, when rolling two dice and getting a sum of 7 which can be achieved either by (1, 6), (2, 5), (3, 4), (4, 3), (3, 4), (5, 2), or (6, 1).

What is a Complementary Event?

Two disjoint events are called complementary events if their union is a complete sample space. For example, in a coin toss one event is getting a head and its complementary event is getting tails.

What is the Difference Between Mutually Exclusive Events and Independent Events?

Those events which can’t occur simultaneously are called Mutually exclusive events whereas independent events are those events where the occurrence of one event doesn’t affect the probability of the other event.

What is the Difference Between Conditional Probability and Unconditional Probability?

When the probability of an event is calculated without any prior knowledge of any event associated with it, then that probability is called unconditional probability whereas conditional probability is the probability of an event occurring given that another event has already occurred.

What are Events in Probability?

In probability, events are the outcomes of an experiment. The probability of an event is the measure of the chance that the event will occur as a result of an experiment.

What is the Difference Between Sample Space and Event?

A sample space is a collection or a set of possible outcomes of a random experiment while an event is the subset of sample space. For example, if a die is rolled, the sample space will be {1, 2, 3, 4, 5, 6} and the event of getting an even number will be {2, 4, 6}.

What is the Probability of an Impossible Event and a Sure Event?

The probability of a sure event is always 1 while the probability of an impossible event is always 0.

What is an Example of an Impossible Event?

An example of an impossible event will be getting a number greater than 6 when a die is rolled.

What is meant by complementary events?

In probability, two events are said to be complementary if one event takes place if and only if the other event does not take place.

What are the different types of events in probability?

The different types of events in probability are complementary events, simple events, compound events, sure events, impossible events, dependent events, independent events, mutually exclusive events, exhaustive events, etc.

Are complementary events mutually exclusive?

Yes, complementary events are mutually exclusive. This represents that the events that are complementary never happen at the same time.

What is meant by an independent event?

In probability, the independent events are the events that do not depend on the occurrence of the other event. In other words, an event which is not affected by the other event is called an independent event.

Why Is This Important?

Understanding these types of events helps in solving probability problems effectively. Whether it’s for CBSE board exams or JEE Mains & Advanced, mastering these concepts will give you a strong foundation for advanced topics.

📥 Download PDF

Students can download detailed notes and solved examples from ANAND CLASSES to practice and excel in the board exams and JEE preparations.