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: In the principle of chance, two phenomena are said to be mutually exclusive if they can not occur together or at the same time. In other words, disjointed events are considered mutually exclusive events. If two occurrences are known as disjoint occurrences, the likelihood of all occurrences happening at the same time is zero.

Complete step-by-step answer:
Given, a die is rolled and there is an event E in which die shows 4 and also there is an event F in which die shows even number and it is being asked whether E and F mutually exclusive events.
As, a die is rolled so the sample space will be,
 $ S = \left\{ {1,2,3,4,5,6} \right\} $
As in the question it is given,
There is an event E in which die shows 4,
So,
E: Die shows 4 $ $
 $ \Rightarrow {\text{E}} = \left\{ 4 \right\} $
Now, also there is an event F in which die shows even numbers.
F: Die shows an even number, possible outcomes are 2, 4, 6.
 $ {\text{F}} = \left\{ {2,4,6} \right\} $
As, for mutually exclusive events $ {\text{E}} \cap {\text{F}} = \phi $ .
Here, in this case,
 $
\Rightarrow {\text{E}} \cap {\text{F}} = \left\{ 4 \right\} \cap \left\{ {2,4,6} \right\} \\
\Rightarrow {\text{E}} \cap {\text{F}} = \left\{ 4 \right\} ;
  $
Therefore, the events E and F are not mutually exclusive events.
So, the correct answer is “the events E and F are not mutually exclusive events”.

Note: A sample space that depends on the experiment can include a variety of findings. If a finite number of results is used, then it is defined as discrete or finite sample spaces. In curly braces "{}" the sample spaces for a random experiment are written. Between the sampling space and the cases, there is a gap. We can get the sample space for rolling a die, S as {1, 2, 3, 4, 5, 6 }, while the case can be written as {1, 3, 5 } representing the set of odd numbers and { 2, 4, 6 } representing the set of even numbers. The results of an experiment are spontaneous, and with certain special experiments, the sample space becomes the universal set.