If we roll a dice and count how many spots it shows, is it possible that the number of spots be even and less than 2 simultaneously? Of course not. If it is an even number, it is impossible that it is less than 2. And if it is less than 2, it is impossible that it is even.

Let’s see another example. If three coins are rolled once simultaneously, can all the coins show the same type of side and there is exactly one tail appearing? No. If all the coins show the same type of side, the event “there is one tail appearing” cannot occur. And if there is one tail appearing, the event “all the coins show the same type of side” cannot occur.

The examples above, where the two events cannot occur simultaneously, illustrate the concept of mutually exclusive events. Two events are * mutually exclusive* (=

*) if they cannot occur simultaneously. In other words, “the events A and B are mutually exclusive” means that if A occurs then B cannot occur and if B occurs then A cannot occur. Statistically speaking, two events A and B are mutually exclusive if . If then A and B are said to be*

**disjoint***, not mutually exclusive.*

**overlapping**Let’s see another example, where A and B are not mutually exclusive. In rolling a dice, the event “the dice shows more than 2 spots” and the event “the dice shows an odd number of spots” are not mutually exclusive. To prove this, let A = the dice shows more than 2 spots and B = the dice shows an odd number of spots. Then A = {3, 4, 5, 6} and B = {1, 3, 5}. Note that 3 and 5 are common elements of A and B, so . Therefore, . We conclude that A and B are not mutually exclusive events. If a 3 shows, then A and B occur simultaneously. Also, if a 5 shows then A and B both occur. In this case, A and B can occur simultaneously.

Different from the latest example above, the first two examples are the ones where the condition is satisfied. In the first case, let A = the number of spots is even and B = the number of spots is less than 2. Then A = {2, 4, 6} and B = {1}. There is not any common element of A and B. So, and conclude that the events are mutually exclusive. They cannot happen simultaneously.

In the second example, let A = all the coins show the same type of side and B = there is exactly one tail appearing. Then, A = {HHH, TTT} and B = {THH, HTH, HHT}. There is no common element of A and B. So, and conclude that the events are mutually exclusive. They cannot occur simultaneously.

If A and B are not mutually exclusive then . Here denotes the probability that A or B occurs. In the case of A and B being mutually exclusive, . As a consequence, .

**Example 1**

In rolling a dice once, what is the probability that the number of spots is even or less than 2?

**Answer**

The sample space is S = {1, 2, 3, 4, 5, 6}. Then,

Let A = the number of spots is even and B = the number of spots is less than 2.

A = {2, 4, 6}

and

B = {1}

and

As , A and B are mutually exclusive and applies.

Therefore, .

So, the probability that the number of spots is even or less than 2 is 0.67.

**Example 2**

In rolling 3 coins at once, find the probability that all the coins show the same type of side or there is exactly one tail appearing.

**Answer**

The sample space is S = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}. Thus,

Let A = all the coins show the same type of side and B = there is exactly one tail appearing

A = {HHH, TTT}

and

B = {THH, HTH, HHT}

and

Since , A and B are mutually exclusive and applies.

Therefore,

So, the probability that all the coins show the same type of side or there is exactly one tail appearing is 0.625.

**Example 3**

In rolling a dice, find the probability that the dice shows more than 2 spots or it shows an odd number of spots.

**Answer**

The sample space is S = {1, 2, 3, 4, 5, 6}. So,

Let A = the dice shows more than 2 spots and B = the dice shows an odd number of spots

A = {3, 4, 5, 6}

and

B = {1, 3, 5}

and

and

By applying the formula , we have:

.

So, the probability that the dice shows more than 2 spots or it shows an odd number of spots is 0.83.

**Example 4**

Two dice are rolled at once. What is the probability that a sum of 8 appears or both dice show the same number of spots.

**Answer**

The sample is S = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}.

Then,

Let A = a sum of 8 appears and B = both dice show the same number of spots

A = {(2,6), (3,5), (4,4), (5,3), (6,2)}

and

B = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}

and

and

By applying the formula , we have:

So, the probability that a sum of 8 appears or both dice show the same number of spots is 0.28.