To determine the amount you should win or lose for any other number to make the game fair, we need to find the probabilities of rolling each number.
When rolling two dice, there are a total of 36 possible outcomes (6 outcomes for the first die multiplied by 6 outcomes for the second die).
Let's calculate the probabilities for each desired outcome:
1. If the dice sum to 9:
- There are four possible outcomes to sum to 9: (3, 6), (4, 5), (5, 4), (6, 3).
- The probability of each of these outcomes is 4/36 = 1/9.
- Since you lose $9 for this outcome, you should lose $9 with a probability of 1/9.
2. If the dice sum to 4 or 6:
- There are three possible outcomes to sum to 4: (1, 3), (2, 2), (3, 1).
- There are five possible outcomes to sum to 6: (1, 5), (2, 4), (3, 3), (4, 2), (5, 1).
- The total number of favorable outcomes is 3 + 5 = 8.
- The probability of each of these outcomes is 8/36 = 2/9.
- Since you win $11 for these outcomes, you should win $11 with a probability of 2/9.
Now, to make the game fair, the expected value of the game should be zero. The expected value is calculated by multiplying each possible outcome by its probability and summing them up. In this case, we have:
(-$9) * (1/9) + (+$11) * (2/9) + x * (p) = 0,
where x represents the amount you should win or lose for any other number, and p represents the probability of any other number turning up.
Let's solve the equation for x:
(-$9) * (1/9) + (+$11) * (2/9) + x * (p) = 0,
(-$9/9) + ($22/9) + x * (p) = 0,
($13/9) + x * (p) = 0,
x * (p) = -($13/9),
x = -($13/9p).
So, you should win or lose -$13/9p for any other number to make the game fair.