Chapter 8, Problem 8.75RE

(a)

To determine

To calculate the probability that the player’s chosen number is a winner.

Answer to Problem 8.75RE

The probability is $0.25$ .

Explanation of Solution

  $\begin{array}{l}\text{Given: 20 of the 80 balls are randomly selected}\text{.}\\ \text{The probability is then:}\\ P\text{=}\frac{\text{favorable outcomes}}{\text{possible outcomes}}\\ =\frac{20}{80}\\ =\frac{1}{4}\\ =0.25\end{array}$

Hence the result.

(b)

To determine

To find out the probability distribution of X .

Answer to Problem 8.75RE

The probability distribution is:

X	$3	$-1
Probability	0.25	0.75

Explanation of Solution

If the player wins the game, then the player wins $\$3$ .

  $\begin{array}{l}P(X=3)=P(\text{Win)}\\ \text{=}0.25\end{array}$

If the player loses the game then the player loses the bet of $\$1$ . Use the complement rule:

  $\begin{array}{l}P(X=1)=P(\text{Loss})\\ =1-P(\text{Win)}\\ =1-0.25\\ =0.75\end{array}$

The table is:

X	$3	$-1
Probability	0.25	0.75

(c)

To determine

To calculate the expected value of X .

Answer to Problem 8.75RE

The expected value is $\$0$ .

Explanation of Solution

The expected value is as:

  $\begin{array}{l}\mu ={\displaystyle \sum x}P(X=x)\\ =\$3\times 0.25+(-\$1)\times 0.75\\ =\$0.75-\$0.75\\ =\$0\end{array}$

Hence the result.