4)) There are two different raffles you can enter. )) In raffle \( A \), one ticket will win a \( \$ 520 \) prize, one ticket will win a \( \$ 500 \) prize, one ticket will win a \( \$ 250 \) prize, one ticket will win a \( \$ 70 \) prize, and the remaining tickets will win nothing. There are 1,000 in the raffle, each costing \( \$ 2 \). ()) Raffle B has 250 tickets. Each ticket costs \( \$ 14 \). One ticket will win a \( \$ 340 \) prize, fourteen tickets will win a \( \$ 220 \) prize, and the remaining tickets will win nothing. ()) Which raffle is a better deal? Raffle A Raffle B

To determine which raffle—**Raffle A** or **Raffle B**—is the better deal, we'll analyze the **expected value** for each raffle. The expected value represents the average amount you can expect to win (or lose) per ticket purchased. ### **Raffle A Details:** - **Total Tickets:** 1,000 - **Ticket Price:** \$2 - **Prizes:** - 1 ticket wins \$520 - 1 ticket wins \$500 - 1 ticket wins \$250 - 1 ticket wins \$70 - **Total Winning Tickets:** 4 **Calculating Expected Value for Raffle A:** \[ \text{Expected Winnings} = \left(\frac{520}{1000}\right) + \left(\frac{500}{1000}\right) + \left(\frac{250}{1000}\right) + \left(\frac{70}{1000}\right) = \frac{1340}{1000} = \$1.34 \] **Net Expected Profit/Loss:** \[ \$1.34 \, (\text{winnings}) - \$2 \, (\text{cost}) = -\$0.66 \text{ per ticket} \] *You can expect to lose 66 cents per ticket on average.* --- ### **Raffle B Details:** - **Total Tickets:** 250 - **Ticket Price:** \$14 - **Prizes:** - 1 ticket wins \$340 - 14 tickets win \$220 each - **Total Winning Tickets:** 15 **Calculating Expected Value for Raffle B:** \[ \text{Expected Winnings} = \left(\frac{340}{250}\right) + \left(14 \times \frac{220}{250}\right) = \frac{340 + 3080}{250} = \frac{3420}{250} = \$13.68 \] **Net Expected Profit/Loss:** \[ \$13.68 \, (\text{winnings}) - \$14 \, (\text{cost}) = -\$0.32 \text{ per ticket} \] *You can expect to lose 32 cents per ticket on average.* --- ### **Comparison:** - **Raffle A:** Expected loss of **\$0.66** per ticket - **Raffle B:** Expected loss of **\$0.32** per ticket **Conclusion:** **Raffle B** is the better deal because it has a **higher expected value** (i.e., a smaller average loss per ticket) compared to Raffle A.