# Mutually Exclusive Events

## What are Mutually Exclusive Events?

In probability theory, two events are said to be mutually exclusive if they do not appear at the same time. Mutually exclusive events are also known as disjoint events. If two events are considered as mutually exclusive, then the probability of both the events appearing at the same time is equal to zero. If X and Y are two events, then the

probability of this is written as:

Probability of mutually exclusive event = P ( X and Y) = 0

In probability, the particular addition rule is justifiable when the two events are mutually exclusive. It is defined as the probability of either event appearing is the sum of the probability of each event appearing. If X and Y are said to be mutually exclusive, then the probability of an event X appearing  or the probability of event Y appearing  is derived  as P(X) + P(X) i.e.,

P (X or Y) = P(X) + P(Y)

### Mutually Exclusive Event Definition

Two events are said to be mutually exclusive when both the events cannot take place simultaneously. Mutually exclusive events always undergo different outcomes. These events are also known as disjoint events. Mutually exclusive events prevent the second event to take place when the first event appears. For example: If you toss a coin, if it gives you “ Head” then it will not give you “ Tail” at the same time. Hence, they are mutually exclusive.

It is quite essential to differentiate between mutually exclusive events and independent events. Independent events are those which do not depend on each other whereas mutually exclusive events are those that cannot appear at the same time.

For example: If we throw a die then event X {1,4} and Y { 2,5,6} are mutually exclusive

Similarly, if we throw a die then X  {1, 4} and Z = {2, 4, 5, 6}  are not mutually exclusive because if 4 appears on the die then it will be favorable to both the events X and Z.

### Mutually exclusive Events Examples:

The given below are some of the mutually exclusive event examples:

• If you toss a coin, if it gives you “ Head” then it will not give you “ Tail” at the same time. Hence, they are mutually exclusive and the probability of getting head and tail at the same time is 0.

• In a pack of 52 cards, drawing red cards and clubs are mutually exclusive events; because all the clubs in a card are black.

• In a  six-sided rolling die, the events 4 and 6 are mutually exclusive, because both the event 4 and 5 cannot occur simultaneously if we throw a single die.

If the event α and β are not mutually exclusive, the probability of getting α and β is derived as:

P ( α or β) = P(α) + P(β) – P( α and β)

### Rules for the probability of the mutually exclusive events

As we know, in probability theory, two events are determined as mutually exclusive if they do not take place at the same time. This can be clearly explained with the following example: a toss of a single coin can either appear head or tail but not both at the same time. While tossing the coin, both the outcomes are collectively exhaustive which suggests that at least one of the results may take place. Hence, these two possibilities collectively exhaust every possibility. MHowever, all mutually exclusive events are not commonly exhaustive. For example: when we throw a six-sided die the items 1 and 3 are mutually exclusive as they cannot appear at the same time but not considered as collectively exhaustive because it can result to different outcomes such as ( 2, 4, 5, 6)

From the above definition of mutually exclusive events, we can conclude certain rules.

Addition Rule – P( X+ Y) = 1

Subtraction rule –  P( XU  Y)’ = 0

Multiplication rule – P( X∩ Y) = 0

There are multiple diversifications also. For example, the coin you flip has both sides either head or tail. No matter the number of times you flip a coin, it will always appear heads for the first time you flip the coin and tail when you flip the coin the second time. If we examine the sample space for this experiment,  it will be either H ( for the first time) and T ( for the second time). Such events are known as “sample events” as they have a single point in the sample space. The events are always mutually exclusive.

### Mutually Exclusive Events Formulas

Here are some of the mutually exclusive events formulas which will help you to solve the questions based on mutually exclusive events probability.

The probability of two events says α and β are mutually exclusive is represented as

1. α and β

The intersection set between α  and β is equal to {null}. Hence P (α  and β) = 0.

It is because when two events cannot appear at the same time, then obviously there will be nothing common in that.

1. α or β

P ( α or β) = P(α) + P(β) – P( α and β)

The probability of the union of two mutually exclusive events is derived by the addition of the probabilities of the events separately.

### Solved Examples

Below you can see the mutually exclusive events examples with solution

1. The probabilities of three mutually exclusive events are given as 1/ 6, 2/3 and 1/4. Prove whether the given statement is correct or not.

Solution: Let the events be X, Y and Z

If the events are mutually exclusive, then X ∩ Y = Θ , Y ∩ Z= Θ and X ∩ Z = Θ

If the conditions given above are satisfied , then P ( X ∪Y ∪ Z) = P(X) + P(Y) + (Z)

Since, P( X ∪ Y ∪ Z) = 13/12 > 1,  the probability values comes within 1

Hence, the statement is incorrect.

1. A box containing 4 bulbs, the probability of having one defected bulb is 0.5 and the probability to have zero defected bulb is 0.4. Calculate the probability of one defected bulb and zero defected bulb.

Solution: Probability of single bulb being defected is P(X) = 0.5

Probability of zero bulbs being defected is P(Y) = 0.4

As there can be either zero defected bulb or 1 defected bulb because these two events cannot occur simultaneously. Hence, they are considered as mutually exclusive.

P( X or Y) = 0.5 + 0.4 = 0.9

1. If P(X) = ⅓ and P(Y) = ⅔. Examine whether

1. X and y are mutually exclusive

2. X and Y are exhaustive

Solution:

The events are said to be mutually exclusive if P( X ∩ Y) = 0

The events are said to be exhaustive if P (X ∪ Y) = 1

P(X) + P(Y) = ⅓ + ⅔

P (X ∪ Y) = P(X) + P(Y) + P( X ∩ Y)

1=1-0

If P (X ∪ Y) is the sample space,  then the above two conditions will be satisfied.

Hence, X and Y are mutually exclusive and exhaustive.

### Quiz Time

1. If one card is drawn from a pack of 52 cards then what is the probability of drawing 5 or a diamond.

1. 2/52

2. 4/52

3. 16/52

4. 26/52

2.  If a single die is rolled down then what is the probability of getting an odd number or a 4.

1. ½

2. 1/6

3.  How many face cards are there in a pack of 52 cards?

1. 3

2. 12

3. 13

4. 52

1. Explain the conditional probability of mutually exclusive events.

Conditional probability is defined as the possibility of an event or outcome appearing, based on the occurrence of a previous event or outcome.

Conditional probability is represented as the probability of an event X, provided that another event Y has appeared. Conditional probability for two independent event Y has given X is represented through the expression P(X|A) and it is stated using the equation.

P(Y|X)= P ( X ∩ Y )/P(X)

In other words, the above equation using multiplication is represented as P ( X ∩Y) = 0

P(Y|X)= 0/P(X)

Hence, the conditional probability formula for the  mutually exclusive event is represented as:

P ( Y | X) = 0

2. Explain the mutually exclusive events in probability.

An event is an outcome or the union of an outcome of an experiment. Consider the situation , we throw a dice . Let A be the event of a perfect square number then A (1,4). Whenever an outcome agrees with the condition given in the event, we say that event has appeared.

The definition of the mutually exclusive event is expanded to two or more events. More than 2 events will be considered as mutually exclusive, if the occurrence of one of these rules out the occurrence of all other events. The events P ( 1,2), Q (3), R(6) are mutually exclusive with respect to the experiment of throwing a die.