Question

What is the difference between events that are mutually exclusive and those that are not mutually exclusive?

Answer 1

Any event can be viewed as a combination of certain elementary events. Thus, an event of rolling a dice on an even number is, actually, a set of three elementary events - rolling numbers 2, 4 and 6.

Now, if two events are considered, each representing a subset of a set of all possible elementary events, they might or might not have certain number of elementary events in common. If they don't, these two events are called mutually exclusive. If they do, they are not mutually exclusive.

An example of mutually exclusive events are `A`=The dice is rolled on even number and `B`=The dice is rolled on odd number.
Obviously, there is no elementary event that simultaneously belongs to events `A` and `B` since a number cannot be even and odd at the same time.

An example of not mutually exclusive events are `X`=The dice is rolled on even number and `Y`=The dice is rolled on a number greater than 3.
Obviously, the elementary events `e_1`=The dice is rolled on number 4 and `e_2`=The dice is rolled on number 6 belong to both events `A` and `B` since numbers 4 and 6 are both even and greater than 3.