The adjusted payout should be $2.78 to make the game fair for Bob.
Determine the probability of drawing a derangement from the jar, and compare it to the expected payoff of $2.50 to determine if the game is fair for Bob.
Calculating the probability of drawing a derangement:
A derangement is a permutation where no element is fixed, meaning no number is drawn in its original position. To calculate the probability of drawing a derangement, consider the first ball drawn. There are 50 possible positions for the first ball, but only 49 are derangements (since drawing the ball in its original position is not a derangement).
For the second ball, there are 49 remaining positions, but only 48 are derangements. Similarly, for each subsequent ball, there are one fewer derangement possibility.
Therefore, the total number of derangements is 50! - 49! + 48! - 47! + ... + 1!
This can be simplified using the formula for the number of derangements:
D_n = n! * (1 - 1/2 + 1/3 - 1/4 + ... + 1/n)
Plugging in n = 50, we get:
D_50 = 50! * (1 - 1/2 + 1/3 - 1/4 + ... + 1/50) ≈ 9.902 * 10^44
Since there are 50! total permutations, the probability of drawing a derangement is:
P(derangement) = D_50 / 50! ≈ 0.198047
Expected payoff:
The expected payoff is the average amount of money Bob can expect to win or lose over many trials. Since Bob pays $1 to play and wins $2.50 with probability P(derangement) and wins $0 otherwise, the expected payoff is:
Expected payoff = -1 + 2.50 * P(derangement) ≈ -1 + 2.50 * 0.198047 ≈ -0.501953
Is the game fair?
A fair game is one where the expected payoff is equal to the amount paid to play. In this case, the expected payoff is negative, indicating that Bob is expected to lose money over time.
To make the game fair, the payout should be adjusted so that the expected payoff is equal to $0.
Adjusted payout = (1 + Expected payoff) / P(derangement) = (1 - 0.501953) / 0.198047 ≈ $2.7809
Therefore, the adjusted payout should be $2.78 to make the game fair for Bob.
Question
Alice proposes to Bob the following game. Bob pays one dollar to play. Fifty balls marked 1,2,....,50 are placed in a big jar, stirred around, and then drawn out one by one by Zori, who is wearing a blindfold. The result is a random permutation ? of the integers 1, 2,...,50. Bob wins with a payout of two dollars and fifty cents if the permutation ? is a derangement, i.e., ? (i)\\eqi for all i = 1, 2,..., n. Is this a fair game for Bob? If not how should the payoff be adjusted to make it fair?