Question

A die is rolled. Let E be the event “die shows \[4\]” and F be the event “die shows even number”. Are E and F mutually exclusive?

Answer 1

Hint:First write down the sample space for a die, recall the outcomes possible for a die. Then write down the possible outcomes for events E and F. To check whether the events E and F are mutually exclusive or not, recall the condition for mutually exclusive events.

Complete step by step solution:
Given, a die is rolled. The event E is when the die shows 4 .

The event F is when the die shows an even number.

First we will write the sample space for the die. By sample space we mean the set of all possible outcomes when the die is rolled. We know a die has six sides having the numbers \[1,\,2,\,3,\,4,\,5\]
and \[6\] on each side. Therefore the sample space for die is
\[S = \left\{ {1,\,2,\,3,\,4,\,5,\,6} \right\}\]

We have the event E when the die shows the number \[4\]. Therefore, event E can be written as,
\[E = \left\{ 4 \right\}\] (i)

We have the event F when the die shows an even number. From the sample space we can see that there are three even numbers \[2,\,4,\,6\]. So these are the possible outcomes for event F. therefore, event F can be written as,

\[F = \left\{ {2,\,4,\,6} \right\}\] (ii)

Now we will find whether E and F are mutually exclusive events. By mutually exclusive events we mean the events cannot occur together or at the same time.
We will now check if there any number common between E and F for which both events can occur at the same time.

Using equations (i) and (ii) we find the intersection of event E and F
\[E \cap F = \left\{ 4 \right\}\]

We get that the number \[4\] is common between both the events. But for the events E and F to be mutually exclusive there should not be any common number between the events as this means both events can occur at the same time.

Therefore, the events E and F are not mutually exclusive events.

Note:There are two important terms that one should remember, these are mutually exclusive events and mutually inclusive events. As we have discussed above mutually exclusive events cannot occur together or are independent of each other, whereas mutually inclusive events are events which can occur at the same time or cannot occur independently. Here if it was asked whether the events E and F are mutually inclusive events and the answer would be yes, they are mutually inclusive events.