Final answer:
The expected amount you win in this dice game is $5.69 per game. After accounting for the $5 entry fee, the expected profit is $0.69 per game.
Step-by-step explanation:
To calculate the expected amount you win at this game, we will first define the possible outcomes and their probabilities. When rolling a pair of dice, there are a total of 36 possible outcomes since each die has 6 faces and the outcome on one die does not affect the outcome on the other. To win $50, you must roll a sum greater than 10, which can only occur with the combinations (5, 6), (6, 5), and (6, 6), resulting in a probability of 3/36 or 1/12.
To win $30 from getting a double, the possible outcomes are (1, 1), (2, 2), (3, 3), (4, 4), and (5, 5) with a probability of 5/36. However, rolling a double six (6, 6) would give you $80, which we already accounted for in the first case, so the probability of winning $30 is actually 4/36, as we exclude the (6, 6) outcome. To win $80 for getting a double six, the probability is 1/36. With this information, we can calculate the expected winnings.
Let's calculate the expected value (EV):
- EV = (1/12 * $50) + (4/36 * $30) + (1/36 * $80) - ($5 * (1 - (3/36 + 4/36)))
- EV = (50/12) + (120/36) + (80/36) - 5 * (1 - 7/36)
- EV = (4.17 + 3.33 + 2.22) - 5 * (29/36)
- EV = $9.72 - $4.03
- EV = $5.69
Therefore, the expected amount you win is $5.69 for each game played before the $5 entry fee. When the cost to play the game is factored in, the expected profit is $0.69 per game.