Mutually Exclusive Events

Introduction: 

In statistics, two or more events are said to be mutually exclusive events if they do not happen or occur at the same time. The tossing of a coin, a left-hand turn or a right-hand turn on the road, a number on a dice, even and odd numbers, and victory and defeat in a game are all examples of mutually exclusive events. In the above examples, we can only get one event at a time, the occurrence of another is not at all possible. 

Define Mutually Exclusive Events:

The term “mutually exclusive events,” is defined as, two or more events that are said to be mutually exclusive if they do not occur simultaneously also known as a “disjoint event.” 

Examples:

1. Tossing of a coin

A coin just has two sides, a head and a tail. So when it is tossed there is the possibility either the tail occurs or the head occurs but the occurrence of both outcomes is not possible at the same time.

2. Even and Odd number

Suppose a bucket has some slips of even and odd numbers, so the drawing of a slip can only results in either an even number or an odd but the occurrence of both numbers at the same time is not possible.

3. Rolling of a Dice

Dice have 6 numbers from 1 to 6. When it is rolled, we can get only one number instead of them all at the same time. 

4. Victory and Defeat in the game

This is a universal fact that at the end of the game one player or a team wins and others lose it. But victory and defeat cannot occur at the same time for the same player or a team.

Rules of Mutually Exclusive Events:

Mutually exclusive events  follow some fundamental probability rules as mentioned below:

If a and b are two mutually exclusive events then

• Addition Rule

          P (a union b) = P (a ꓴ b) 

          P (a ꓴ b) = P(a) + P(b)

• Subtraction Rule

          P (a ꓴ b)’ = 0

• Multiplication Rule

          P (a ∩ b) = 0

Venn Diagram Representation:

Non-Mutual Exclusive Events:

Two circles in the rectangular portion represent the events. The common portion of both events depicts the intersection of events, these events are non-mutual exclusive events.

The formula for non-mutual exclusive events.

If a and b are two non-mutual exclusive events then

P (a ꓴ b) = P(a) + P(b) – P (a ∩ b)

Mutually Exclusive Events:

Two circles in the rectangular portion represent the events. As there is no common portion of both events, these events are mutually exclusive.

The formula for non-mutual exclusive events.

If a and b are two mutually exclusive events then

P (a ꓴ b) = P(a) + P(b)

Solved Examples:

Example 1:

 A die is taken and rolled once. An event is defined as M1, a set of possible outcomes where the number on the face of the die is less than 4, and event M2 is the set of possible outcomes where the number on the face of the die is 4 and greater than 4. Are event1 M1 and M2 mutually exclusive?

Solution1:

Given details: M1 contains the number less than 4 on the face of the die and M2 contains the number 4 or greater than 4 face of the die.

M1 = {1,2,3}

M2 = {4,5,6}

When a dice is rolled, dice can show either a number less than 4 or 4 and greater than 4 so M1 and M2 are mutually exclusive events.

Example 2:

 A die is taken and rolled once. Event M1 is the set of all possible outcomes where the number on the face of the die is odd and event M2 is the set of possible outcomes where the number on the face of the die is less than 5. Are events M1 and M2 mutually exclusive?

Solution2:

Given details: M1 contains the odd face of the die and M2 contains the number less than 5.

E1 = {1,3,5}

E2 = {1,2,3,4}

1 and 3 common elements, so M1 and M2 are mutually exclusive events.

Conclusion:

Several times it becomes important to know whether the events are mutually exclusive or not. Two or more events are said to be mutually exclusive events if they cannot occur at the same time. If a and b are two mutually exclusive events then they will not be sharing any outcomes i.e., P (a and b) = 0. Knowing the type of events influences the calculation of the probability of one or the other occurring. If two events are mutually exclusive then, the probability of either occurring is the sum of each occurring.