Question

You roll a die with the sample space s=1,2,3,4,5,6. you define A as (1,2,4) B as (2,3,4,5,6) C as (3,4) and D as (2,4,5). Determine which of the following events are exhaustive and/or mutually exhaustive.

Exhaustive Mutually exclusive
A and B
A and C
A and D
Band C

Answer 1

A and B are exhaustive.

Explanation:

Given

`A = \{1,2,4\}`

`B = \{2,3,4,5,6\}`

`C =\{3,4\}`

`D = \{2,4,5\}`

Solving (a): The mutually exclusive events

These are events that have no common or mutual elements

Events A to D are not mutually exclusive because each of the events have at least 1 common element with one another.

Solving (b): Exhaustive events.

Two events X and Y are said to be exhaustive if:

`S = P(X\ n\ Y)`

i.e. if the sample space equals the intersection of X and Y

For events A to D, we have:

`A\ n\ B = \{1,2,3,4,5,6\}`

and the sample space is:

`S = \{1,2,3,4,5,6\}`

By comparison;

`A\ n\ B = S`

Hence, A and B are exhaustive.