Questions: You are holding a raffle to fundraise for college scholarships. There will be one grand prize winner for 1000 and 3 consolation prize winners for 90. The cost for people to purchase a single ticket will be 6. If you sell 1148 tickets, then the expected profit per ticket sold for your fundraiser will be 4.89373. Round this expected value to the nearest cent (two decimal places).

You are holding a raffle to fundraise for college scholarships. There will be one grand prize winner for 1000 and 3 consolation prize winners for 90. The cost for people to purchase a single ticket will be 6. If you sell 1148 tickets, then the expected profit per ticket sold for your fundraiser will be 4.89373. Round this expected value to the nearest cent (two decimal places).

Transcript text: You are holding a raffle to fundraise for college scholarships. There will be one grand prize winner for $\$ 1000$ and 3 consolation prize winners for $\$ 90$. The cost for people to purchase a single ticket will be $\$ 6$. If you sell 1148 tickets, then the expected profit per ticket sold for your fundraiser will be $\$ 4.89373$. Round this expected value to the nearest cent (two decimal places).

Solution

Solution Steps

To find the expected profit per ticket, we need to calculate the total expected payout and subtract it from the total revenue, then divide by the number of tickets sold. The total revenue is the number of tickets sold multiplied by the price per ticket. The expected payout is the sum of the probabilities of winning each prize multiplied by the prize amount. Finally, we round the expected profit per ticket to the nearest cent.

Step 1: Calculate Total Revenue

The total revenue from selling tickets is given by the formula: \[ \text{Total Revenue} = \text{Number of Tickets} \times \text{Ticket Price} = 1148 \times 6 = 6888 \]

Step 2: Calculate Probabilities

The probability of winning the grand prize is: \[ P(\text{Grand Prize}) = \frac{1}{\text{Number of Tickets}} = \frac{1}{1148} \approx 0.0008711 \] The probability of winning a consolation prize is: \[ P(\text{Consolation Prize}) = \frac{\text{Number of Consolation Winners}}{\text{Number of Tickets}} = \frac{3}{1148} \approx 0.0026132 \]

Step 3: Calculate Expected Payout

The expected payout can be calculated as follows: \[ \text{Expected Payout} = P(\text{Grand Prize}) \times \text{Grand Prize} + P(\text{Consolation Prize}) \times \text{Consolation Prize} \times \text{Number of Consolation Winners} \] Substituting the values: \[ \text{Expected Payout} = 0.0008711 \times 1000 + 0.0026132 \times 90 \times 3 \approx 1.5767 \]

Step 4: Calculate Expected Profit per Ticket

The expected profit per ticket is calculated using the formula: \[ \text{Expected Profit per Ticket} = \frac{\text{Total Revenue} - (\text{Expected Payout} \times \text{Number of Tickets})}{\text{Number of Tickets}} \] Substituting the values: \[ \text{Expected Profit per Ticket} = \frac{6888 - (1.5767 \times 1148)}{1148} \approx 4.4233 \]

Step 5: Round to the Nearest Cent

Rounding the expected profit per ticket to the nearest cent gives: \[ \text{Expected Profit per Ticket (Rounded)} \approx 4.42 \]