Mutually Exclusive Events

Mutually exclusive events are those that cannot occur at the same time; if one event happens, it eliminates the possibility of the other event occurring. For example, in a cricket match between India and Pakistan, only one team can win. If Pakistan wins, India loses, and vice versa. Therefore, "India winning" and "Pakistan winning" are mutually exclusive events, as the occurrence of one excludes the other.

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.

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.

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.

Note: We can also say that independent events are never mutually exclusive events.

Learn More about Dependent and Independent 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  the probability of B is P(B) then the probability of happening both events together is, P(A∩B) = 0

Then the probability of any one event is:

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

Here, we define ∩ 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 "∩" 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 A∩B and A∩B = {2, 4, 6}

Union of Sets
The symbol which defines the union is "∩" 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 events A and B as,

P ( A∩B) = 0...(i)

We know that,
P (A U B) = P(A) + P(B) - P ( A∩B)
But if A and B are mutually exclusive events then, by (i) we get,
P (A U B) = P(A) + P(B)

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(H∩T) = 0
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. 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 that 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)’ = 0
• Multiplication Rule: P (A ∩ B) = 0

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

Conditional Probability for Mutually Exclusive Events

Conditional probability is the probability of event A occurring given that 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 ∩ B)/P(A)

Now if A and B are two mutually exclusive events then by using the multiplication rule P (A ∩ B) = 0.
P(B|A) = 0/P(A) = 0

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

P(B | A) = 0

Read More,

• Bayes’ Theorem
• Binomial Theorem
• Probability Distribution

Solved Examples

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

Given,
P(A ∪ B) = 0.51 and P(A) = 0.20

We know that,
P(A ∪ 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 ∪ 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 ∪ B) = 0.65, and A and B are mutually exclusive events then find P(A).
Solution:

Given,
P(A ∪ B) = 0.65 and P(B) = 0.35

We know that,
P(A ∪ 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