Chapter 12.4, Problem 16PPS

a.

To determine

Find the expected number of stripes.

Answer to Problem 16PPS

  $2.1875$

Explanation of Solution

Given information:

In a rack of $16$ billiard balls, there are $9$ different colors, including the black eight ball and the white cue ball. Of the remaining $14$ balls, $7$ are striped and $7$ are solid. If $5$ balls are randomly selected.

Calculation:

Out of $16$ balls $7$ are striped, so the probability of striped balls,

  $\begin{array}{l}P\left(strip\right)=\frac{7}{16}\\ so,\exp ectednumberofstripballis,\\ E\left(strip\right)=5\cdot P\left(strip\right)\\ =5\times \frac{7}{16}\\ =2.1875\end{array}$

Hence, expected value is $2.1875$ .

b.

To determine

Find the expected number of white balls.

Answer to Problem 16PPS

  $0.25$

Explanation of Solution

Given information:

In a rack of $16$ billiard balls, there are $9$ different colors, including the black eight ball and the white cue ball. Of the remaining $14$ balls, $7$ are striped and $7$ are solid. If $4$ balls are randomly selected.

Calculation:

  $\begin{array}{l}p\left(white\right)=\frac{1}{16}\\ sothe\exp ectednumberofwhiteballis,\\ E\left(white\right)=4\times p\left(white\right)\\ =4\times \frac{1}{16}\\ =0.25\end{array}$

Hence, expected value is $0.25$ .

c.

To determine

Find the expected value of randomly selected ball.

Answer to Problem 16PPS

  $0.96875$

Explanation of Solution

Given information:

In a rack of $16$ billiard balls, there are $9$ different colors, including the black eight ball and the white cue ball. Of the remaining $14$ balls, $7$ are striped and $7$ are solid.

If the value of the cue ball is $0$ and the other ball are numbered $1-15$ .

Calculation:

In a rack of $16$ billiard balls, and each ball equally likely so the probability is,

  $\begin{array}{l}p\left(0\right)=p\left(1\right)=p\left(2\right)\mathrm{...}=p\left(15\right)=\frac{1}{16}\\ sothe\exp ectednumberofrandomballis,\\ E\left(ball\right)={\displaystyle \sum _{i=0}^{15}i\times p\left(i\right)}\\ =\frac{15}{16}\\ =0.96875\end{array}$

Hence, expected value is $0.96875$ .