To find the expected profit of the game, we need to calculate the probability of each outcome and multiply it by the corresponding profit/loss amount. Let's calculate the expected profit step by step:
1. Calculate the probabilities:
- Probability of getting a sum greater than 10: There are 6 possible outcomes (5, 6), (6, 5), (5, 6), (6, 6), (5, 6), (6, 5) out of 36 total outcomes when rolling two dice. Therefore, the probability is 6/36 = 1/6.
- Probability of getting a double: There are 6 possible outcomes (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6) out of 36 total outcomes. Therefore, the probability is 6/36 = 1/6.
- Probability of getting a double and a sum greater than 10: There is only one possible outcome (6, 6) out of 36 total outcomes. Therefore, the probability is 1/36.
2. Calculate the profit/loss for each outcome:
- If a sum greater than 10 is rolled: Profit = $50 - $5 (cost to play) = $45.
- If a double is rolled: Profit = $30 - $5 (cost to play) = $25.
- If a double and a sum greater than 10 are rolled: Profit = $80 - $5 (cost to play) = $75.
- For all other outcomes: Profit = -$5 (cost to play).
3. Calculate the expected profit:
Expected Profit = (Probability of outcome 1 * Profit of outcome 1) + (Probability of outcome 2 * Profit of outcome 2) + ...
Expected Profit = (1/6 * $45) + (1/6 * $25) + (1/36 * $75) + (29/36 * -$5)
Expected Profit = $7.50 + $4.17 + $2.08 - $5.00
Expected Profit = $8.75 - $5.00
Expected Profit = $3.75
Therefore, the expected profit of this game is $3.75.