1. The problem statement, all variables and given/known data 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 (let's call it s) of the integers 1, 2, . . . , 50. Bob wins with a payout of two dollars and fifty cents if the permutation s is a derangement, i.e., s(i) =/= i 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? 2. Relevant equations Derangement formula: ∑(k=from 0 to n) [(-1)^k * C(n,k) * (n-k)!] 3. The attempt at a solution I got this far: Probability that any given random permutation is a derangement: (num of all derangements) / (num of all permutations) = ∑(k=from 0 to n) [(-1)^k * C(n,k) * (n-k)!] / 50! ≈ 1/e Then, computing the expected win/loss, given that he gains $2.50 (for a net profit of $2.50 - $1 - $1.50) if he wins, and is out $1.00 if he loses: 1.5(1/e) - 1(1-1/e) ≈ -0.08. (so on average, he'll lose roughly 8 cents per play) Solving for the adjusted payment amount so that he'll break even in the long run... (p-1)(1/e) - 1(1-1/e = 0 Solving for p, we get: p ≈ e First, is the above correct? Right now I settled for an approximate answer (e) as the derangement equation seemed too messy to compute an exact answer. Is there a simpler way to get an exact answer? Thanks.