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?
2. if a person rolls doubles when he tosses two dice, he wins $5. the cost to play the game is $1. is this game fair?

Explanation:

Step1: Calculate the total number of bills

The total number of bills is \(10 + 5+3 + 1+1=20\)

Step2: Calculate the probability of each bill

• Probability of \(\$1\) bill: \(P(1)=\frac{10}{20}\)
• Probability of \(\$2\) bill: \(P(2)=\frac{5}{20}\)
• Probability of \(\$5\) bill: \(P(5)=\frac{3}{20}\)
• Probability of \(\$10\) bill: \(P(10)=\frac{1}{20}\)
• Probability of \(\$100\) bill: \(P(100)=\frac{1}{20}\)

Step3: Calculate the expected value of the bill

The expected value \(E(X)\) of the bill is:

$$
LATEXBLOCK0
$$

Step4: Calculate the expected value of the game

The cost to play the game is \(\$20\). The expected value of the game \(E\) is \(E = 7.25-20=- 12.75\)

Answer:

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