Question

the world famous gambler from philadelphia, señor rick, proposes the following game of chance. you roll a fair die. if you roll a 1, then señor rick pays you $25. if you roll a 2, señor rick pays you $5. if you roll a 3, you win nothing. if you roll a 4 or a 5, you must pay señor rick $10, and if you roll a 6, you must pay señor rick $15. is señor rick proposing a fair game?

Explanation:

Step1: Calculate probabilities

A fair die has 6 possible outcomes.

• Probability of rolling a 1: \(P(1)=\frac{1}{6}\)
• Probability of rolling a 2: \(P(2)=\frac{1}{6}\)
• Probability of rolling a 3: \(P(3)=\frac{1}{6}\)
• Probability of rolling a 4: \(P(4)=\frac{1}{6}\)
• Probability of rolling a 5: \(P(5)=\frac{1}{6}\)
• Probability of rolling a 6: \(P(6)=\frac{1}{6}\)

Step2: Calculate expected value

Let \(X\) be the random variable representing the gain or loss.

• If roll 1: \(X = 25\)
• If roll 2: \(X = 5\)
• If roll 3: \(X = 0\)
• If roll 4 or 5: \(X=- 10\) (since pay \(10\))
• If roll 6: \(X=- 15\) (since pay \(15\))

The expected value formula is \(E(X)=\sum_{i}x_{i}P(x_{i})\)

$$
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Since the expected value \(E(X)\approx - 0.83\) (a loss in the long - run), the game is not fair.

Answer:

No. You lose $\$0.83$ in the long run.