Chapter 18, Problem 42E

(a)

To determine

To create a probability model for this game.

Explanation of Solution

It is given in the question that you pay $\$10$ and roll a die, if you get a $6$ you win $\$50$ but if not, you get to roll again. If you get a $6$ this time you get your $\$10$ back. Thus, let X be the amount of money won. Then the probability model for this game will be as:

(b)

To determine

To find the expected value and the standard deviation of your prospective winnings.

Answer to Problem 42E

The expected value is $\$-0.28$ and the standard deviation is $\$18.33$ .

Explanation of Solution

It is given in the question that you pay $\$10$ and roll a die, if you get a $6$ you win $\$50$ but if not, you get to roll again. If you get a $6$ this time you get your $\$10$ back. Thus, let X be the amount of money won. Thus, the expected value and the standard deviation of your prospective winnings is calculated as:

  $\begin{array}{l}\mu =E(X)\\ =40\times \frac{1}{6}+0\times \frac{5}{36}-10\times \frac{25}{36}\\ =\$-0.28\\ {\sigma }^2=Var(X)\\ $ ={(40-(-0.28))}^2\times \frac{1}{6}+{(0-(-0.28))}^2\times \frac{5}{36}+{(-10-(-0.28))}^2\times \frac{25}{36}$=336.034\\ \sigma =SD(X)\\ =\sqrt{Var(X)}\\ =\sqrt{336.034}\\ =\$18.33\end{array}$

(c)

To determine

To find the expected value and the standard deviation of your average winnings if you play this game five times.

Answer to Problem 42E

The expected value is $\$-1.40$ and the standard deviation is $\$40.99$ .

Explanation of Solution

It is given in the question that you pay $\$10$ and roll a die, if you get a $6$ you win $\$50$ but if not, you get to roll again. If you get a $6$ this time you get your $\$10$ back. Thus, let X be the amount of money won. Thus, the expected value and the standard deviation of your prospective winnings if you play this game five times is calculated as:

  $\begin{array}{l}\mu =E(X+X+X+X+X)\\ =5E(X)\\ =5(-0.28)\\ =\$-1.40\\ \sigma =SD(X+X+X+X+X)\\ =\sqrt{5Var(X)}\\ =\sqrt{5(336.034)}\\ =\$40.99\end{array}$

(d)

To determine

To find out what is the probability the person running the game makes a profit if $100$ people play this game.

Answer to Problem 42E

The probability that the person running the game makes a profit is approximately $0.561$ .

Explanation of Solution

It is given in the question that you pay $\$10$ and roll a die, if you get a $6$ you win $\$50$ but if not, you get to roll again. If you get a $6$ this time you get your $\$10$ back. Thus, let X be the amount of money won. Thus, in order for the person running the game to make a profit the average winnings of the $100$ people must be less than dollar zero. The expected value is $\$-0.28$ and the standard deviation is $\$18.33$ . Thus, rolling a die is random and the outcomes are mutually independent so the Central Limit Theorem guarantees that the sampling distribution model is Normal with the mean and the standard deviation as:

  $\begin{array}{l}{\mu }_{\overline{x}}=\$-0.28\\ {\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}\\ =\frac{18.33}{\sqrt{100}}\\ =\$1.83\end{array}$

Thus, the z -score will be as:

  $\begin{array}{l}z=\frac{0-(-0.28)}{1.83}\\ =0.153\end{array}$

Thus, using the graphing utility to find the probability the person running the game makes a profit if $100$ people play this game is calculated as:

  $\begin{array}{l}P(z<0.153)=normalcdf(-E99,0.153,0,1)\\ =0.561\end{array}$

Thus, the probability that the person running the game makes a profit is approximately $0.561$ .