Question 14.8: Selecting a Monetary Bill A box contains five $1 bills, thre......
Selecting a Monetary Bill
A box contains five $1 bills, three $5 bills, and two $10 bills. A person selects a bill at random. What is the expected value of the bill? Perform the experiment 25 times.
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