Chapter 3.4, Problem 36E

a.

To determine

Find the expected value and standard deviation of total winnings when a gambler bets $3 on a single round.

Answer to Problem 36E

The expected value and standard deviation of total winnings when a gambler bets $3 on a single round are –0.081 and 2.9989, respectively.

Explanation of Solution

From the given information at a university, there are 37 slots with 18 red slots, 18 black slots, and 1 green slot. The gambler can place their bets on red or black; if the ball lands on their color, they double their money and if the ball lands on other color, they lose their money.

There are a total of 37 outcomes in that there are18 chances that the ball lands on a red, and there are 18 chances that the ball lands on the black color. There is one chance that the ball lands on the green color.

The expected value is calculated as follows:

$\begin{array}{c}E\left(X\right)={\displaystyle \sum _{i=1}^n{X}_i\times P\left(X_i\right)}\\ E\left(\text{winnings}\right)=\left(\$3\times \frac{18}{37}\right)-\left(\$3\times \frac{18}{37}\right)-\left(\$3\times \frac{1}{37}\right)\\ =-\frac{3}{37}\\ \approx -0.081\end{array}$

Thus, the expected value is –0.081.

Standard deviation:

$\begin{array}{c}E\left(X^2\right)={\displaystyle \sum _{i=1}^{}{X}_i{}^2\times }P\left(X_i\right)\\ =3^2\left(\frac{18}{37}\right)+\left(-3^2\right)\left(\frac{18}{37}\right)+\left(-3^2\right)\left(\frac{1}{37}\right)\\ =9\\ SD\left(X\right)=\sqrt{E\left(X^2\right)-{\left[E\left(X\right)\right]}^2}\\ =\sqrt{9-{\left(\frac{-3}{37}\right)}^2}\\ =2.9989\end{array}$

Thus, standard deviation is 2.9989.

b.

To determine

Find the expected value and standard deviation of total winnings when a gambler bets $1 in three different rounds.

Answer to Problem 36E

The expected value and standard deviation of total winnings when a gambler bets $1 in three different rounds are –0.081 and 173, respectively.

Explanation of Solution

From the given information at a university, there are 37 slots with 18 red slots, 18 black slots, and 1 green slot. The gambler can place their bets on red or black; if the ball lands on their color, they double their money and if the ball lands on other color, they lose their money.

There are a total of 37 outcomes in that there are18 chances that the ball lands on the red and there are 18 chances that ball lands on the black color. There is one chance that the ball lands on the green color.

The expected value is calculated as follows:

$\begin{array}{c}E\left(X\right)={\displaystyle \sum _{i=1}^n{X}_i\times P\left(X_i\right)}\\ E\left(\text{winnings}\right)=\left(\$1\times \frac{18}{37}\right)-\left(\$1\times \frac{18}{37}\right)-\left(\$1\times \frac{1}{37}\right)\\ =-\frac{1}{37}\\ \approx -0.027\end{array}$

The expected value of the total winning for a single round is $–0.027.

The expected value for three rounds is calculated as follows:

$\begin{array}{c}E\left(Winningforthreerounds\right)=-0.027\times 3\\ =-0.081\end{array}$

Thus, the expected value is $–0.081.

Standard deviation:

$\begin{array}{c}E\left(X^2\right)={\displaystyle \sum _{i=1}^{}{X}_i{}^2\times }P\left(X_i\right)\\ =1^2\left(\frac{18}{37}\right)+\left(-1^2\right)\left(\frac{18}{37}\right)+\left(-1^2\right)\left(\frac{1}{37}\right)\\ =1\end{array}$

For three rounds $E\left(X^2\right)=3$

$\begin{array}{c}SD\left(X\right)=\sqrt{E\left(X^2\right)-{\left[E\left(X\right)\right]}^2}\\ =\sqrt{3-{\left(\frac{-3}{37}\right)}^2}\\ =1.73\end{array}$

Thus, standard deviation is 1.73.

c.

To determine

Compare the results obtained from Part (a) and Part (b) and describe the riskiness of two games.

Answer to Problem 36E

The expected values for both the games are the same but Part (b) game has a less standard deviation than Part (a).

The bet of $3 on a single round is riskier than bet of $1 in three different rounds.

Explanation of Solution

The results obtained in Part(a) and Part(b) are given as follows:

The expected value and standard deviation of total winnings when a gambler bets $3 on a single round are –0.081 and 2.9989, respectively.

The expected value and standard deviation of total winnings when a gambler bets $1 in three different rounds are –0.081 and 1.73, respectively.

The expected values for both the games are the same, but Part (b) games have a less standard deviation than Part (a).

Thus, a bet of $3 on a single round is riskier than a bet of $1 in three different rounds.