91Ó°ÊÓ

First, we need to list all possible outcomes of rolling a fair die and flipping a fair coin and determine the respective winnings based on the given rules: 1. Die-roll: 1, Coin-flip: Heads - Winnings: 2(1) = 2 2. Die-roll: 1, Coin-flip: Tails - Winnings: 1/2(1) = 0.5 3. Die-roll: 2, Coin-flip: Heads - Winnings: 2(2) = 4 4. Die-roll: 2, Coin-flip: Tails - Winnings: 1/2(2) = 1 5. Die-roll: 3, Coin-flip: Heads - Winnings: 2(3) = 6 6. Die-roll: 3, Coin-flip: Tails - Winnings: 1/2(3) = 1.5 7. Die-roll: 4, Coin-flip: Heads - Winnings: 2(4) = 8 8. Die-roll: 4, Coin-flip: Tails - Winnings: 1/2(4) = 2 9. Die-roll: 5, Coin-flip: Heads - Winnings: 2(5) = 10 10. Die-roll: 5, Coin-flip: Tails - Winnings: 1/2(5) = 2.5 11. Die-roll: 6, Coin-flip: Heads - Winnings: 2(6) = 12 12. Die-roll: 6, Coin-flip: Tails - Winnings: 1/2(6) = 3

Next, we need to calculate the probability of each of the listed outcomes. Since there are 6 sides on a fair die and 2 sides on a fair coin, the total number of possible outcomes is 6 x 2 = 12. Therefore, the probability of one specific outcome is 1/12. Now, we can calculate the expected value of the winnings for each of these outcomes by multiplying the probability of each outcome by the respective winnings: 1. \(E(1, H) = \frac{1}{12} * 2 = \frac{1}{6}\) 2. \(E(1, T) = \frac{1}{12} * 0.5 = \frac{1}{24}\) 3. \(E(2, H) = \frac{1}{12} * 4 = \frac{1}{3}\) 4. \(E(2, T) = \frac{1}{12} * 1 = \frac{1}{12}\) 5. \(E(3, H) = \frac{1}{12} * 6 = \frac{1}{2}\) 6. \(E(3, T) = \frac{1}{12} * 1.5 = \frac{1}{8}\) 7. \(E(4, H) = \frac{1}{12} * 8 = \frac{2}{3}\) 8. \(E(4, T) = \frac{1}{12} * 2 = \frac{1}{6}\) 9. \(E(5, H) = \frac{1}{12} * 10 = \frac{5}{6}\) 10. \(E(5, T) = \frac{1}{12} * 2.5 = \frac{5}{24}\) 11. \(E(6, H) = \frac{1}{12} * 12 = 1\) 12. \(E(6, T) = \frac{1}{12} * 3 = \frac{1}{4}\)

Finally, we need to add up the expected values from each outcome to find the overall expected winnings for the player: \(E(total) = E(1, H) + E(1, T) + E(2, H) + E(2, T) + E(3, H) + E(3, T) + E(4, H) + E(4, T) + E(5, H) + E(5, T) + E(6, H) + E(6, T)\) \(E(total) = \frac{1}{6} + \frac{1}{24} + \frac{1}{3} + \frac{1}{12} + \frac{1}{2} + \frac{1}{8} + \frac{2}{3} + \frac{1}{6} + \frac{5}{6} + \frac{5}{24} + 1 + \frac{1}{4}\) \(E(total) = \frac{25}{12} = 2.0833\) The expected winnings for the player when throwing a fair die and flipping a fair coin simultaneously is approximately 2.0833 units.