Some events of a probability experiment can occur simultaneously and some events cannot occur simultaneously.

For example, consider the experiment of throwing die. Let us define the event A as getting an odd number and the event B as getting an event number and the third event C as getting a multiple of 3. The outcomes of the three events can be listed as follows:
A = {1, 3, 5} B = {2, 4, 6} and C = {3, 6}
Events A and C have the outcome 3 as common, while B and C have the outcome 6 in common. The events A and B do not share any common outcomes. Suppose the outcome turns out to be 3, we conclude that both the events A and C have occurred. And for the outcome 6, we say that both the events B and C have occurred. But there is no possibility of saying that both the events A and B have occurred. A and B are hence the mutually exclusive events. This against events A and C or the events B and C are not mutually exclusive and hence mutually inclusive events.

## Mutually Exclusive Events Definition

Two events are called mutually exclusive events if they cannot occur at the same.  This means they do not share common outcomes.As we saw above when you throw a die, the events of getting an odd number and getting an even number are mutually exclusive.  Similarly when you toss a coin the events of getting a Head and getting the Tail are mutually exclusive.

An event and its complement are mutually exclusive.

A number of events E1, E2, E,3 .........En are called mutually exclusive if the occurrence of one event denies the occurrence of other events. if these events together form the sample space, then they are called mutually exclusive and exhaustive.

When a fair die is thrown, the events of getting 1, 2, 3, 4 , 5 and 6 are all mutually exclusive and exhaustive events.

In fact, the sample space for any experiment consists of mutually exclusive simple events.

The event A ∩ B is a null set when A and B are mutually exclusive. Hence P(A ∩ B) = 0.Thus the probability of either A or B occurs is the sum of their probabilities.

Addition law for the union of two mutually exclusive events
P(A or B) = P(A) + P(B).

## Mutually Exclusive Events Examples

1. In a multiple choice question, getting the answer correct and getting the wrong answer are mutually exclusive events.

2. At a political rally, there are 25 Republicans, 30 Democrats and 10 independents. If a person is selected at random find the probability that she or he is a Republican or a Democrat.
As the person chosen cannot be both Republican and Democrat, we can use the addition rule for Mutually exclusive events here.
P( Republican or Democrat) = P(Republican) + P(Democrat)
= $\frac{25}{65}$ + $\frac{30}{65}$ = $\frac{55}{65}$ = $\frac{11}{13}$.

3. For an online degree program, the available instructors for various subjects are as follows:
 Subject Number of Instructors Math 10 Physics 6 Chemistry 5 Biology 5 Statistics 4

If an instructor is selected as the Lead, find the probability that the person selected teaches either Math or Stats assuming that each person teaches only one subject. Since each person teaches only one subject, we can apply the addition rule for mutually exclusive events here.
P(Math or Stats) = P(Math) + P(Stats)
= $\frac{10}{30}$ + $\frac{4}{30}$ = $\frac{14}{30}$ = $\frac{7}{15}$.

4. Identify whether the given events are mutually exclusive or not.
(a) Selecting a student who is either a sophomore or doing Math major.
(b) Selecting a student who is either a sophomore or a Freshman.
(a) These events are not mutually exclusive, as a sophomore doing Math major can be picked.
(b) These two events are mutually exclusive as a student cannot be both a sophomore and a Freshman.

### Compound Events

 Inclusion Exclusion Examples Compound Event Dependent Events Types of Events Sample Spaces Vector Space Sample Variance of the Sample Means Sampling Distribution of Sample Mean Population and Sampling Binomial Distribution Sample Problems Hypothesis Testing Sample Problems Compounded Interest
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