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# What Are Mutually Exclusive Events – Definition, Formula & Examples

January 23, 2023

Probability is the study of computing a numerical value for the likelihood of happening or not happening events. These events can be mutually exclusive events or non-mutually exclusive events. For example, the events that a coin lands on its head or tail are mutually exclusive events. On the other hand, getting an odd number or a prime number when a die is rolled are non-mutually exclusive events.

Let’s understand what are mutually exclusive events and how to check whether the events are mutually exclusive with examples.

## What are Mutually Exclusive Events?

Two events associated with a random experiment are said to be mutually exclusive if both cannot occur simultaneously in the same trial. Thus, mutually-exclusive events are also known as disjoint events, i.e., the events with no common element.

Let’s consider a random experiment of rolling a die. Further, let there are two events say $\text{E}_1$ getting an odd number and $\text{E}_2$ getting an even number. The events $\text{E}_1$ and $\text{E}_2$ are mutually exclusive events.

Then the event $\text{E}_1$ can be written as $\{1, 3, 5 \}$ and the event $\text{E}_2$ can be written as $\{2, 4, 6 \}$. You can see that there is no element in common between  $\text{E}_1$ and $\text{E}_2$, or we can say that $\text{E}_1$ and $\text{E}_2$ are disjoint.

Now, let’s consider two more events say $\text{E}_3$ getting an odd number and $\text{E}_4$ getting a prime number. The events $\text{E}_3$ and $\text{E}_4$ are non-mutually exclusive events(or not mutually exclusive events).

The event $\text{E}_3$ can be written as $\{1, 3, 5 \}$ and the event $\text{E}_4$ can be written as $\{2, 3, 5 \}$. You can see that there is one element in common between  $\text{E}_1$ and $\text{E}_2$, and it is $3$. Therefore, we can say that $\text{E}_3$ and $\text{E}_4$ are non-disjoint (or not disjoint).

Hence, we can say that if $\text{A}$ and $\text{B}$ are two events, then $\text{A} \text{ OR } \text{B}$ or $\left( \text{A} \cup \text{B} \right)$ denotes the event of the occurrence of at least one of the events $\text{A}$ or $\text{B}$.

$\text{A}$ and $\text{B}$ or $\left(\text{A} \cap \text{B} \right)$ is the event of the occurrence of both events $\text{A}$ and $\text{B}$.

Note:

• If $\text{A}$ and $\text{B}$ are mutually exclusive events, then $\text{P} \left(\text{A} \cap \text{B} \right) = 0$.
• If $\text{A}$ and $\text{B}$ are not mutually exclusive events, then $\text{P} \left(\text{A} \cap \text{B} \right) \ne 0$.

## How Do You Calculate Mutually Exclusive Events?

Mutually exclusive events are events that cannot occur or happen at the same time, i.e., the occurrence of mutually exclusive events at the same time is $0$. If $\text{A}$ and $\text{B}$ are two mutually exclusive events, the probability of them both happening together is $\text{P} \left(\text{A AND B} \right) = 0$.

The formula for calculating the probability of two mutually exclusive events is given as $\text{P} (\text{A OR B}) = \text{P}(\text{A}) + \text{P}(\text{B})$.

## Conclusion

Two events are said to be mutually exclusive if they cannot occur at the same time or simultaneously. And conversely, non-mutually exclusive events are the ones that can happen simultaneously. If the intersection of two events is an empty event (number of elements is 0), then we say that the two events are mutually exclusive events.