Question 14.8: Selecting a Monetary Bill A box contains five $1 bills, thre......

Step 1 List all possible outcomes. They are $\$ 1, \$ 5$, and $\$ 10$.

Step 2 Assign the probabilities to each outcome:

$P(\$ 1)=\frac{5}{10} \quad P(\$ 5)=\frac{3}{10} \quad P(\$ 10)=\frac{2}{10}$

Step 3 Set up a correspondence between the random numbers and the outcomes. Use random numbers 1 through 5 to represent a $\$ 1$ bill being selected, 6 through 8 to represent a $\$ 5$ bill being selected, and 9 and 0 to represent a $\$ 10$ bill being selected.

Step 4 Select 25 random numbers and tally the results.

$\begin{array}{|l|l|}\hline \text{Number }& \text{Results }(\$) \\\hline 45829 & 1,1,5,1,10 \\25646 & 1,1,5,1,5 \\91803 & 10,1,5,10,1 \\84060 & 5,1,10,5,10 \\96943 & 10,5,10,1,1 \\\hline\end{array}$

Step 5 Compute the average:

$\bar{X}=\frac{\Sigma X}{n}=\frac{\$ 1+\$ 1+\$ 5+\cdots+\$ 1}{25}=\frac{\$ 116}{25}=\$ 4.64$

Hence, the average (expected value) is $\$ 4.64$.

Recall that using the expected value formula $E(X)=\Sigma[X \cdot P(X)]$ gives a theoretical average of

$E(X)=\Sigma[X \cdot P(X)]=(0.5)(\$ 1)+(0.3)(\$ 5)+(0.2)(\$ 10)=\$ 4.00$