Let's solve the problem step by step: ### Problem Recap: - You have **$1 million**. - If you answer **correctly (probability = 10% = 0.1)**, you win **$3 million**. - If you answer **incorrectly (probability = 90% = 0.9)**, your prize is **reduced to $750,000**. - Ignore your current winnings for the expected payoff calculation. --- ### Step 1: Calculate the Expected Payoff **Expected Value (EV) = (Probability of Correct) × (Payoff if Correct) + (Probability of Incorrect) × (Payoff if Incorrect)** But, we want the **expected change** from playing the final round, so compare to not playing (where you just keep $1M): - If you **play**: EV = 0.1 × $3,000,000 + 0.9 × $750,000 EV = $300,000 + $675,000 EV = $975,000 - If you **don’t play**: You keep $1,000,000. So the **expected payoff from playing** (relative to just keeping $1M) is: **Expected Payoff = $975,000 - $1,000,000 = -$25,000** --- ### Step 2: Should You Play? Since the expected payoff is **negative** (–$25,000), you **should not play** the final round. --- ### Step 3: Minimum Probability for Zero Expected Value Let \( p \) be the probability of answering correctly. Set up the expected value equation: \[ EV = p \times \$3,000,000 + (1-p) \times \$750,000 \] Set the expected payoff equal to your current winnings ($1,000,000): \[ p \times \$3,000,000 + (1-p) \times \$750,000 = \$1,000,000 \] Solve for \( p \): \[ 3,000,000p + 750,000 - 750,000p = 1,000,000 \] \[ (3,000,000 - 750,000)p = 1,000,000 - 750,000 \] \[ 2,250,000p = 250,000 \] \[ p = \frac{250,000}{2,250,000} = \frac{1}{9} \approx 0.1111 \text{ or } 11.11\% \] --- ### **Final Answers:** 1. **Expected payoff** from playing the final round is: \(\boxed{-\$25,000}\) 2. **You should not play** the final round (since payoff is negative). 3. **Lowest probability to break even** (expected value of zero) is: \(\boxed{11.11\%}\) --- **Summary Table:** | Step | Value | |-------------------------------|----------------------| | Expected Payoff | -$25,000 | | Play the Final Round? | No | | Minimum Probability (Break-Even) | 11.11% |