Question

Horse racing. Each part of this exercise contains events from Exercise 5.54. In each case, decide whether the event are mutually exclusive.

Part (a) A and B

Part (a) B and C

Part (a) A, B and C

Part (a) A,B and D

Part (a) A, B, C and D

Short answer

Part (a) The events A and B are mutually exclusive

Part (b) The events B and C are mutually exclusive

Part (c) The events A,B and C are not mutually exclusive

Part (d) The events A,B and D are mutually exclusive

Part (e) The events A,B,C and D are not mutually exclusive

Step-by-step solution

Part (a) Step 1. Given information.

$$\begin{gathered} A=\{3,4\} \\ B=\{6,7,8\} \\ C=\{1,2,3\} \\ D=\{2,5\} \end{gathered}$$

Part (a) Step 2., The events A and B are mutually exclusive 

Because they have no shared outcomes, events A and B are mutually exclusive.

$A\cap B=\varphi$

Part (b) Step 1., The events B and C are mutually exclusive 

Because they have no shared outcomes, events B and C are mutually exclusive.

$B\cap C=\varphi$

Part (c) Step 1., A, B, and C are not mutually exclusive.

A, B, and C are not mutually exclusive. Because there are two of them with similar results, they are mutually exclusive.

$A\cap C=3$

Part (d) Step 1. A, B, and D are mutually exclusive events.

$$\begin{gathered} A\cap B=\varphi \\ B\cap D=\varphi \\ A\cap D=\varphi \end{gathered}$$

Therefore, A, B, and D are mutually exclusive events.

Part (d) Step 1. A, B, C, and D are not mutually exclusive events.

$A\cap C=3$

Therefore A, B, C, and D are not mutually exclusive events.