For a fundraiser, there is a raffle with 1,000 tickets. One ticket will win a \( \$ 290 \) prize, even tickets will win a \( \$ 230 \) prize, nine tickets will win a \( \$ 90 \) prize, and the rest will win othing. If you have a ticket, what is the expected payoff?

To find the expected payoff for a ticket in the raffle, we need to calculate the expected value based on the probabilities of winning each prize and the corresponding amounts. ### Known Conditions: 1. Total tickets: \( N = 1000 \) 2. Prizes: - 1 ticket wins \( \$290 \) - 500 tickets (even tickets) win \( \$230 \) - 9 tickets win \( \$90 \) - The rest win nothing. ### Step-by-Step Calculation: 1. **Calculate the probabilities of winning each prize:** - Probability of winning \( \$290 \): \[ P(290) = \frac{1}{1000} \] - Probability of winning \( \$230 \): \[ P(230) = \frac{500}{1000} = \frac{1}{2} \] - Probability of winning \( \$90 \): \[ P(90) = \frac{9}{1000} \] - Probability of winning nothing: \[ P(0) = 1 - P(290) - P(230) - P(90) = 1 - \frac{1}{1000} - \frac{500}{1000} - \frac{9}{1000} = \frac{490}{1000} \] 2. **Calculate the expected value (EV):** The expected value is calculated as follows: \[ EV = (P(290) \times 290) + (P(230) \times 230) + (P(90) \times 90) + (P(0) \times 0) \] Substituting the probabilities and prizes: \[ EV = \left(\frac{1}{1000} \times 290\right) + \left(\frac{500}{1000} \times 230\right) + \left(\frac{9}{1000} \times 90\right) + \left(\frac{490}{1000} \times 0\right) \] 3. **Perform the calculations:** - For \( \$290 \): \[ \frac{1}{1000} \times 290 = 0.290 \] - For \( \$230 \): \[ \frac{500}{1000} \times 230 = 115 \] - For \( \$90 \): \[ \frac{9}{1000} \times 90 = 0.81 \] - For nothing: \[ \frac{490}{1000} \times 0 = 0 \] 4. **Sum the expected values:** \[ EV = 0.290 + 115 + 0.81 + 0 = 116.1 \] ### Final Answer: The expected payoff for a ticket in the raffle is \( \$116.10 \).