Chapter 13.4, Problem 32E

To determine

To find whether the pair of events is mutually exclusive or mutually inclusive and find the probability.

Answer to Problem 32E

Mutually exclusive, $\frac{55}{221}$

Explanation of Solution

Given:

The probability that the two drawn cards from a standard deck will both be aces or both will be black.

Calculation:

The standard cards contains $26$ red cards, $4$ aces ( $2$ red and $2$ black). In total there are $52$ cards

Let $A$ be the event of selecting both aces.

Let $B$ be the event of selecting both black.

Sample space $52$

The events are mutually inclusive as there can be a black ace.

Therefore probability of event $A$ is $P(A)=\frac{4}{52}\cdot \frac{3}{51}=\frac{12}{2652}$

And probability of event $B$ is $P(B)=\frac{26}{52}\cdot \frac{25}{51}=\frac{650}{2652}$

Hence,

  $\begin{array}{l}P(\text{A or B})=P(A)+P(B)-P(A\text{ and }B\text{)}\\ P(\text{A Or B})=\frac{12}{2652}+\frac{650}{2652}-\frac{2}{2652}=\frac{660}{2652}=\frac{55}{221}\end{array}$