# Mutually Exclusive Events

Last Updated : 12 Apr, 2024

We define mutually exclusive events as events that can never happen simultaneously, i.e. happening an event rules out the possibility of happening the other event. Suppose a cricket match between India and Pakistan can result in the winning of any one team and the loss of the other team both teams can never win the match simultaneously, i.e. if Pakistan wins the match then India definitely loses the match and if India wins the match Pakistan definitely loses the matches thus, we can say Winning of India and Winning of Pakistan both are mutually exclusive events. And occurring one event definitely rules the probability of the other event.

## Mutually Exclusive Events Definition

Two events are said to be mutually exclusive if they can’t occur simultaneously. In other words, mutually exclusive events are called disjoint events. If two events are considered disjoint events, then the probability of both events occurring at the same time is zero.

### Examples of Mutually Exclusive Events

Some examples of mutually exclusive events are,

• Tossing a coin we either get a head or a tail. Head and tail cannot appear simultaneously. Therefore, the occurrence of a head or a tail is two mutually exclusive events.
• In throwing a die all the 6 faces numbered 1 to 6 are mutually exclusive as if any one of these faces comes on the top, the possibility of others in the same trial is ruled out.

### Dependent and Independent Events

An event in probability falls under two categories,

• Dependent Events
• Independent Events

Dependent Events: We define two events as dependent events if the occurrence of one event changes the probability of another event. For example, tossing a coin if we get a tail we can never get a head in the same trial.

Independent Events: We define two events as independent events if the occurrence of one event does not change the probability of another event. For example, taking a card from a well-shuffled deck can be either a face card or a black card or both here all the events are independent events. We can also say that independent events are never mutually exclusive events.

## How to Calculate Mutually Exclusive Events?

We know that mutually exclusive events are events that can not occur simultaneously and if we take two events A and B as mutually exclusive events and the probability of A is P(A) and Â the probability of B is P(B) then the probability of happening both events together is,

Then the probability of occurring any one event is,

P(AUB) = P(A) or P(B) = P(A) + P(B)

Here, we define âˆ© the symbol as the intersection of the set and the U symbol as the union of the set. Before proceeding further let’s learn about the Intersection of the set and the Union of the set.

Intersection of Sets

The symbol which defines the intersection is “âˆ©” it is also called “AND”. We define Intersection as the values that are contained in both sets, i.e.

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

Then A intersection B is represented as Aâˆ©B and Aâˆ©B = {2, 4, 6}

Union of Sets

The symbol which defines the union is “âˆ©” it is also called “OR”. We define Union as all the values contained in both sets, i.e.

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

Then A union B is represented as AUB and AUB = {1, 2, 3, 4, 5, 6}

## Probability of Mutually Exclusive Events OR Disjoint Events

We know that mutually exclusive events can never happen simultaneously and thus the probability of Mutually Exclusive Events is always zero. We define the probability of the mutually exclusive event A and B as,

We know that,

P (A U B) = P(A) + P(B) – P ( Aâˆ©B)

But if A and B are mutually exclusive events then, by (i) we get,

P (A U B) = P(A) + P(B)

For Example: In a coin toss probability of getting a head is P(H) and the probability of getting a tail is P(T) and both getting head and getting a tale is a mutually exclusive events,

Then,

• P(H) = 0.50
• P(T) = 0.50

P(HUT) = P(H) + P(T) = 0.50 + 0.50 = 1

## Mutually Exclusive Events Venn Diagram

Venn diagrams are widely used to represent mutually exclusive events. We know that if we represent a set using a circle in the Venn diagram then in mutually exclusive events we get the Venn diagram in which we have nothing in common between the two sets as shown in the image below:

### Non-Mutually Exclusive Events

In non-mutually exclusive events we get the Venn diagram in which we some common parts between the two sets as shown in the image below:

### Do Mutually Exclusive Events Add up to 1?

We know that it is not possible for mutually exclusive events to occur simultaneously. And the probability of any event can never be greater than one and as we already know mutually exclusive events are dependent events and thus their probability is never greater than one.

Now, the probability of mutually exclusive events can add up to 1 only if the events are exhaustive, i.e. at least one of the events is true.

### Real-Life Examples of Mutually Exclusive Events

Mutually Exclusive events or disjoint events are events in probability theory that never occur simultaneously. We can understand it as suppose we have a box containing 5 red balls and 5 blue balls then if we draw a ball it can either be red or blue but can never be both. Thus, these are mutually exclusive events but, if we number each ball from 1 to 5 respectively and then draw a ball and look for either an even-numbered ball or red color. Now in this case it can occur that the ball is even-numbered and red in colour and thus it is a non-mutually exclusive event.

Some other real-life examples of mutually exclusive events are, while throwing a die getting any two numbers simultaneously is a mutually exclusive event. Getting head and tail simultaneously while tossing a coin is a mutually exclusive event.

## Mutually Exclusive Events Probability Rules

We use the following rules for simplifying the mutually exclusive events such as if A and b are two mutually exclusive events then,

• Addition Rule: P (A + B) = 1
• Subtraction Rule: P (A U B)â€™ = 0
• Multiplication Rule: P (A âˆ© B) = 0

Simple events which only have one [possible outcome are always mutually exclusive to other simple events.

## Conditional Probability for Mutually Exclusive Events

We define conditional probability as the probability of event A when another event B has already occurred. We define the conditional probability of event B when event A has already occurred as, P( B|A)

We can calculate its value as,

P(B|A) = P (A âˆ© B)/P(A)…(ii)

Now if A and B are two mutually exclusive events then by using the multiplication rule P (A âˆ© B) = 0 in eq (i)

P(B|A) = 0/P(A) = 0

Thus, the formula for conditional probability for mutually exclusive events is,

P (B | A) = 0

## Summary – Mutually Exclusive Events

The articles explain the concept of mutually exclusive events in probability, which are events that cannot occur at the same time. For instance, the outcome of a coin toss being either heads or tails is mutually exclusive because both outcomes cannot happen simultaneously. These events have a combined probability of zero when trying to occur together.

Furthermore, the distinction between mutually exclusive and non-mutually exclusive events is highlighted through their representation in Venn diagrams. Mutually exclusive events are represented without overlapping circles, indicating no common outcomes. In contrast, non-mutually exclusive events show overlaps in their Venn diagrams, depicting possible simultaneous occurrences.

## Solved Examples

Example 1: If P(A) = 0.20, P(A âˆª B) = 0.51, and A and B are mutually exclusive events then find P(B).

Solution:

Given,

P(A âˆª B) = 0.51 and P(A) = 0.20

We know that,

P(A âˆª B) = P(A) + P(B)

0.51 = 0.20 + P(B)

P(B) = 0.51 – 0.20 = 0.31

Thus, P(B) is 0.31

Example 2: In a deck of 52 cards find the probability of getting either an even card or a face card.

Solution:

Probability of getting a even card P(A) = 5/13

Probability of getting a face card P(B) = 3/13

We know that P(A) and P(B) are two mutually exclusive events, then

P(A âˆª B) = P(A) + P(B)

= 5/13 + 3/13

= 8/13

Thus, the probability of getting either a even card or a face card is 8/13.

Example 3: If P(B) = 0.35, P(A âˆª B) = 0.65, and A and B are mutually exclusive events then find P(A).

Solution:

Given,

P(A âˆª B) = 0.65 and P(B) = 0.35

We know that,

P(A âˆª B) = P(A) + P(B)

0.65 = 0.65 + P(A)

P(A) = 0.65 – 0.35 = 0.30

Thus, P(A) is 0.30

## FAQs on Mutually Exclusive Events

### What are Mutually Exclusive Events in Probability?

In probability, we define mutually exclusive events as events that can not occur simultaneously. That is if one event happens then it is impossible for another event to happen.

### What are examples of Mutually Exclusive Events?

Mutually exclusive events are events that can not occur simultaneously, i.e.

• If we toss a coin, we only get either H or T but we can not get both H and T and hence it is a mutually exclusive event.
• Similarly, if we roll a die it only shows any one of the six faces on the top, and 1 and 2 both can never come on top at once, hence they are mutually exclusive events.

### What is the Formula for Mutually Exclusive Events?

Let A and B be two mutually exclusive events and the probability of occurring A is P(A) and the probability of occurring B is P(B), then

P(AUB) = P(A) + P(B)

### How to know if A and B are Mutually Exclusive Events?

Two mutually exclusive events are events that can not occur simultaneously. We can say that if A and b are two mutually exclusive events then the intersection of their probability is 0. i.e.

### Are Mutually Exclusive Events Independent or Dependent?

Two mutually exclusive events are always dependent in nature. If one event happens, it affects the probability of happening of the other event.

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