Question

games & expectation

1. a box contains ten $1 bills, five $2 bills, three $5 bills, one $10 bill, and one $100 bill. a person is charged $20 to select one bill. find the expected value for this game. is this game fair?

Explanation:

Step1: Calculate total number of bills

Total number of bills \(n = 10 + 5+3 + 1+1=20\)

Step2: Calculate probabilities and values

• For \(\$1\) bill: Probability \(P(1)=\frac{10}{20}\), net value \(V(1)=1 - 20=- 19\)
• For \(\$2\) bill: Probability \(P(2)=\frac{5}{20}\), net value \(V(2)=2 - 20=-18\)
• For \(\$5\) bill: Probability \(P(5)=\frac{3}{20}\), net value \(V(5)=5 - 20=-15\)
• For \(\$10\) bill: Probability \(P(10)=\frac{1}{20}\), net value \(V(10)=10 - 20=-10\)
• For \(\$100\) bill: Probability \(P(100)=\frac{1}{20}\), net value \(V(100)=100 - 20 = 80\)

Step3: Calculate expected value

Expected value \(E=\sum_{i}P(i)\times V(i)\)

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Answer:

The expected value of the game is \(-\$12.75\). Since the expected value is not \(0\), the game is not fair.