Final answer:
To make the game fair, a player should lose approximately $2.32 if any other number than 4, 8, or 11 turns up on the roll of a pair of fair dice.
Step-by-step explanation:
To determine how much a player should win or lose if any other number turns up for the game to be fair, we need to calculate the expected value of the game as it currently stands and then solve for the unknown amount that would make the expected value equal to zero, indicating a fair game. A fair die has six faces, so when rolling two dice, there are 36 possible outcomes. The sums that result in a win or loss for the current game are:
- Sum of 4 losing $8: There are 3 ways to roll a sum of 4 (1,3), (2,2), (3,1).
- Sum of 8 winning $12: There are 5 ways to roll a sum of 8 (2,6), (3,5), (4,4), (5,3), (6,2).
- Sum of 11 winning $12: There are 2 ways to roll a sum of 11 (5,6), (6,5).
To calculate the expected value (EV), we multiply each outcome's value by its probability and sum those products:
EV = (3/36 * -$8) + (5/36 * $12) + (2/36 * $12) + (P * X)
Where P is the probability of any other sum occurring, and X is the amount won or lost on these occasions.
The probability of any other sum occurring is 1 - (Probability of a sum of 4, 8, or 11) = 1 - (3/36 + 5/36 + 2/36) = 26/36.
To set the EV to 0 for fairness:
- Calculate the current EV without considering X.
- Divide the negative of that result by the probability of the other sums (26/36).
- This will give you the value of X to make the game fair.
Doing the math, the fair amount X can be solved as follows:
0 = (3/36 * -$8) + (5/36 * $12) + (2/36 * $12) + (26/36 * X)
0 = (-$24 + $60 + $24)/36 + (26/36 * X)
0 = $60/36 + (26/36 * X)
0 = $1.67 + (26/36 * X)
-$1.67 = (26/36 * X)
X = -$1.67 * (36/26)
The fair amount X is approximately -$2.32. This means to make the game fair, a player should lose $2.32 if any other number turns up.