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 $\text{P} (\text{A OR B}) = \text{P}(\text{A}) + \text{P}(\text{B})$.

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

While calculating mutually exclusive events, we use two important relationships between two sets. These are the intersection of sets and the union of sets.

**Intersection of Sets:**The symbol used for the intersection is ‘$\cap$’ and ‘$\text{AND}$’ is also used. If two sets are there say for example; $\text{A} = {1, 2, 3}$ and $\text{B} = {2, 3, 4}$. Then $\text{A}$ intersection $\text{B}$ is represented as $\text{A} \cap \text{B}$. Therefore, in this case $\text{A} \cap \text{B} = {2, 3}$.**Union of Sets:**The symbol used for the union is “$\cup$ ” and “$\text{OR}$” is also used. If two sets are there say for example $\text{A} = {1, 2, 3}$ and $\text{B} = {2, 3, 4}$.. Then $\text{A}$ union $\text{B}$ is represented as $\text{A} \cup \text{B}$. Therefore, in this case $\text{A} \cup \text{B} = {1, 2, 3, 4}$.

## Examples of Mutually Exclusive Events

Some examples of mutually exclusive events are:

- When a coin is tossed, the event of getting head and tail are mutually exclusive. Because the probability of getting head and tail simultaneously is $0$.
- In a six-sided die, the events “$2$” and “$6$” are mutually exclusive. We cannot get both events $2$ and $6$ at the same time when a die is thrown.
- In a deck of $52$ cards, drawing a red card and drawing a spade are mutually exclusive events because all the spades are black.

## Examples of Non-Mutually Exclusive Events

Some examples of mutually exclusive events are:

- In a six-sided die, the events “$2$” and “a prime number” are non-mutually exclusive. It’s because when we throw a die and get $2$ then we get a prime number also. ($2$ is a prime number).
- In a deck of $52$ cards, drawing a red card and drawing a diamond are non-mutually exclusive events because a red card can be a diamond also. (In a deck of cards diamonds and hearts are red cards)

## Representation of Mutually Exclusive Events

We can use Venn diagrams to show mutually exclusive events. The figures shown below indicate mutually exclusive events and events that are not mutually exclusive events or non-mutually exclusive events. Note that there is no common element in mutually exclusive events.

## Rules for Mutually Exclusive Events

We learned above that two events are mutually exclusive or disjoint if they do not occur at the same time. This can be clearly seen in the set of results of a single coin toss, which can end in either heads or tails, but not for both. While tossing the coin, both outcomes are collectively exhaustive, which suggests that at least one of the consequences must happen, so these two possibilities collectively exhaust all the possibilities.

Though, not all mutually exclusive events are commonly exhaustive. For example, the outcomes $2$ and $6$ of a six-sided die, when we throw it, are mutually exclusive (both $2$ and $6$ cannot come as result at the same time) but not collectively exhaustive (it can result in distinct outcomes such as $1$, $3$, $4$, $5$).

There are some related to mutually exclusive events. These rules are

**Addition Rule:**$\text{P} (\text{A} + \text{B}) = 1$**Subtraction Rule:**$\text{P} (\text{A} \cup \text{B})’ = 0$**Multiplication Rule:**$\text{P} (\text{A} \cap \text{B}) = 0$

## Conditional Probability for Mutually Exclusive Events

Conditional probability is the probability of an event A, given that another event B has occurred. Conditional probability for two independent events $\text{A}$ and $\text{B}$ is represented as $\text{P}(\text{A} | \text{B})$ and it means the probability of an event $\text{A}$ given that $\text{B}$ has already occurred.

Mathematically, the conditional probability is represented as $\text{P}(\text{A}|{B})= \text{P} \frac{(\text{A} \cap \text{B})}{\text{P}(\text{B})}$.

Now, in the case of two mutually exclusive events $\text{A}$ and $\text{B}$, $\text{P} (\text{A} \cap \text{B}) = 0$.

Therefore, $\text{P}(\text{A} | \text{B})= \frac{0}{\text{P}(\text{B})}$.

Thus, the conditional probability formula for mutually exclusive events is given by $\text{P} (\text{A} | \text{B}) = 0$ and similarly, $\text{P} (\text{B} | \text{A}) = 0$ (i.e., the probability of occurrence of an event $\text{B}$ is $0$, given that the event $\text{A}$ has already occurred.

## Examples on Mutually Exclusive Events

**Example 1:** What is the probability of a die showing a number $2$ or a number $6$?

Let a die showing a number $2$ be an event $\text{A}$.

And let a die showing a number $6$ be an event $\text{B}$.

The two events $\text{A}$ and $\text{B}$ are mutually exclusive events, therefore the probability of event $\text{A}$ or event $\text{B}$ is given by $\text{P}(\text{A} \cup \text{B}) = \text{P}(\text{A}) + \text{P}(\text{B})$

$=\frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}$.

Therefore, the probability of a die showing a number $2$ or a number $6$ is $\frac{1}{3}$.

**Example 2:** Three coins are tossed simultaneously. Let $\text{A}$ be the event of receiving at least $2$ heads. Similarly, $\text{B}$ denotes the event of getting no heads, and $\text{C}$ is the event of getting heads on the second coin. Which of these is mutually exclusive?

Let’s first create a sample space for each event.

For the event $\text{A}$ we have to get at least two heads. Therefore, $A = {\text{HHT}, \text{HTH}, \text{THH}, \text{HHH}}$.

This set $\text{A}$ has $4$ elements i.e. $n(\text{A}) = 4$.

Similarly, for event $\text{B}$, the sample is $\text{B} = {\text{TTT}}$ and $n(\text{B}) = 1$.

And, in the same way, for event $\text{C}$, the sample space is $\text{C} = {\text{THT}, \text{HHH}, \text{HHT}, \text{THH}}$ and $n(\text{C}) = 4$.

Thus the events $\text{B}$ & $\text{B}$ and $\text{A}$ & $\text{B}$ are mutually exclusive since they have nothing in their intersection.

**Example 3: ** For two events $\text{A}$ and $\text{B}$, if $\text{P} (\text{A}) = \frac{2}{3}$, $\text{P} (\text{B}) = \frac{1}{2}$ and $\text{P} (\text{A} \cup \text{B}) = \frac{5}{6}$ then check whether the events $\text{A}$ and $\text{B}$ are mutually exclusive events or not.

$\text{P} (\text{A}) = \frac{2}{3}$

$\text{P} (\text{B}) = \frac{1}{2}$

$\text{P} (\text{A} \cup \text{B}) = \frac{5}{6}$

According to the addition rule of probability, we have $\text{P}(\text{A} \cup \text{B}) = \text{P}(\text{A}) + \text{P}(\text{B}) – \text{P}(\text{A} \cap \text{B})$.

Substituting the respective values, we get $\frac{5}{6} = \frac{2}{3} + \frac{1}{2} – \text{P}(\text{A} \cap \text{B})$

$=>\frac{5}{6} = \frac{5}{6} – \text{P}(\text{A} \cap \text{B})$

$=> \text{P}(\text{A} \cap \text{B}) = 0$

Therefore, the events $\text{A}$ and $\text{B}$ are mutually exclusive events.

**Example 4:** A card is drawn at random from a well-shuffled deck of $52$ cards. Find the probability that the card drawn is a jack or a queen.

Let $\text{A}$ be the event of drawing a jack.

And let $\text{B}$ be the event of drawing a queen.

Therefore, $\text{P}(\text{A}) = \frac{1}{13}$ and $\text{P}(\text{B}) = \frac{1}{13}$.

Since the events $\text{A}$ and $\text{B}$ are mutually exclusive events, therefore, the probability of an event $\text{A}$ or an event $\text{A}$ is given by $\text{A} \cup \text{B} = \text{A} + \text{B} = \frac{1}{13} + \frac{1}{13} = \frac{2}{13}$.

Thus, the probability that the card drawn is a jack or a queen is $\frac{2}{13}$.

## Practice Problems

- A die is cast once. What is the probability of getting either a $6$ or a $3$?
- Three faces of a die are painted white such that only the odd number of faces remain visible. The die is cast. What is the probability of getting a $1$ or a $3$ or a $5$?
- In a box containing $4$ bulbs, the probability of having one defective bulb is $0.5$ and the probability to have zero defective bulbs is $0.4$. Calculate the probability of one defective bulb and zero defective bulbs.
- If one card is drawn from a pack of $52$ cards then what is the probability of drawing $5$ or a diamond?
- If a single die is rolled down then what is the probability of getting an odd number or a $4$?

## FAQs

### What is mutually exclusive with example?

Mutually exclusive is a term that describes two or more events that cannot happen simultaneously. It is commonly used to describe a situation where the occurrence of one outcome supersedes the other. For example, getting a head and a tail simultaneously in a single throw of a coin are mutually exclusive events.

### What is non-mutually exclusive with example?

Non-mutually exclusive is a term that describes two or more events that can happen simultaneously. It is commonly used to describe a situation where the occurrence of one outcome along with the other. For example, getting an odd number and a prime number simultaneously in a single roll of a die are non-mutually exclusive events as the event of getting $3$ is both an odd number and a prime number.

### How do you know if an event is mutually exclusive?

Two events are mutually exclusive if they cannot occur at the same time, i.e., the two events are disjoint. If two events are disjoint, then the probability of them both occurring at the same time is $0$. Mathematically, for two events $\text{A}$ and $\text{B}$ to be mutually exclusive $\text{A} \cap \text{B} = 0$.

### What is the rule for mutually exclusive?

Two events are said to be mutually exclusive if they cannot occur at the same time or simultaneously. In other words, mutually exclusive events are called disjoint events. Therefore, the rule for mutually exclusive events $\text{A}$ and $\text{B}$ is $\text{A} \cap \text{B} = 0$.

### How can 3 events be mutually exclusive?

Three events $\text{A}$, $\text{B}$, and $\text{C}$ are mutually exclusive (also called pairwise disjoint) if and only if $\text{P}(\text{A} \cap \text{B}) = 0$, $\text{P}(\text{A} \cap \text{C}) = 0$, and $\text{P}(\text{B} \cap \text{C}) = 0$ It’s trivial to show that this implies that $\text{P}(\text{A} \cap \text{B} \cap \text{C}) = 0$.

## 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.