Chapter 4, Problem 1E

To determine

Calculate the expected payoff and its variance.

Explanation of Solution

The expected payoff is the average, which can be calculated as follows:

$\text{Expected payoff}=\frac{\text{Sum of the earnings}}{\text{Total number of people}}$

The total number of people is 10. Among them, 5 of them earn zero, four of them earn $100, and one loses $100. Thus, the total earnings will be $300. Thus, expected payoff can be calculated as follows:

$\begin{array}{c}\text{Expected payoff}=\frac{\text{300}}{10}\\ =30\end{array}$

Thus, the expected payoff is 30.

The variance can be calculated using the formula given below:

$\text{Variance}={{\displaystyle \sum \text{P}\times \left(\text{Earnings}-\text{Expected payoff}\right)}}^2$

The variance can be calculated as follows:

$\begin{array}{c}\text{Variance}=\left(0.5\times {\left(0-30\right)}^2\right)+\left(0.4\times {\left(100-30\right)}^2\right)+\left(0.1\times {\left(-100-30\right)}^2\right)\\ =450+1960+1690\\ =4100\end{array}$

Thus, the variance is 4100.