In a dice game, you win if the two dice come up 11.
Otherwise, you lose $2.
What should be the profit for winning to make this game fair?
(Give an exact answer. Use symbolic notation and fractions where needed.)
To make the game fair, we need to compute the expected value of the game and set it to zero. Let x be the profit for winning. The probability of winning (rolling a sum of 11) is 2/36 = 1/18, since there are two combinations out of the 36 total possible dice combinations (5 and 6 or 6 and 5). The probability of losing (not rolling a sum of 11) is 17/18. So we have:
Expected value = (1/18)x - (17/18)(2) = 0.
Solving for x:
(1/18)x = (17/18)(2)
x = (17/1)(2)
x = 34
So, the profit for winning should be $34 to make the game fair.