A raffle offers a first prize of \( \$ 500 \) and 3 second prizes of \( \$ 75 \) each. One ticket costs \( \$ 1 \), and 500 tickets are sold. Find the expected payback for a person who buys 1 ticket. Is this a fair game?

To find the expected payback for a person who buys 1 ticket, we need to calculate the expected value of the winnings. Let's denote: - \( P_1 \) as the probability of winning the first prize, - \( P_2 \) as the probability of winning a second prize, - \( E \) as the expected payback. The total number of tickets sold is 500, and the total number of prizes is 1 first prize and 3 second prizes. The probability of winning the first prize is: \[ P_1 = \frac{1}{500} \] The probability of winning a second prize is: \[ P_2 = \frac{3}{500} \] The expected payback for a person who buys 1 ticket is calculated as follows: \[ E = P_1 \times \text{Value of First Prize} + P_2 \times \text{Value of Second Prize} \] The value of the first prize is $500, and the value of each second prize is $75. Now, let's calculate the expected payback. Calculate the value by following steps: - step0: Calculate: \(\frac{1}{500}\times 500+\frac{3}{500}\times 75\) - step1: Reduce the numbers: \(1+\frac{3}{500}\times 75\) - step2: Multiply the numbers: \(1+\frac{9}{20}\) - step3: Reduce fractions to a common denominator: \(\frac{20}{20}+\frac{9}{20}\) - step4: Transform the expression: \(\frac{20+9}{20}\) - step5: Add the numbers: \(\frac{29}{20}\) The expected payback for a person who buys 1 ticket is $1.45. To determine if this is a fair game, we compare the expected payback to the cost of buying a ticket. Since the expected payback is $1.45 and the cost of buying a ticket is $1, the expected payback is greater than the cost of the ticket. Therefore, this is not a fair game.