1)) There is a raffle with 250 tickets. One ticket will win a \( \$ 160 \) prize, one ticket will win \( \$ 30 \) prize, and the rest will win nothing. If you have a ticket, what is the expected payoff

To determine the expected payoff for holding one raffle ticket, we'll analyze the possible outcomes and their associated probabilities. ### Given: - **Total tickets:** 250 - **Prizes:** - **\$160** prize: 1 winning ticket - **\$30** prize: 1 winning ticket - **No prize:** 248 tickets ### Steps to Calculate Expected Payoff: 1. **Determine the Probability of Each Outcome:** - **Winning \$160:** \( \frac{1}{250} \) - **Winning \$30:** \( \frac{1}{250} \) - **Winning Nothing:** \( \frac{248}{250} \) 2. **Calculate the Expected Value (E):** The expected value is the sum of each outcome multiplied by its probability. \[ E = \left( \frac{1}{250} \times \$160 \right) + \left( \frac{1}{250} \times \$30 \right) + \left( \frac{248}{250} \times \$0 \right) \] Simplifying: \[ E = \frac{160 + 30}{250} = \frac{190}{250} = \$0.76 \] ### **Conclusion:** The **expected payoff** for holding one raffle ticket is **\$0.76**. This means that, on average, you'd expect to receive 76 cents per ticket in the long run. However, it's essential to remember that this is a statistical average; in reality, each ticket either wins \$160, \$30, or nothing.