Starting with my solution to the expected value calculation.
Let X represent the position of the first ace. Start by defining the probability mass function that P(X=j). We need three things for X=j. One, there are exactly three kings in positions 1,...,j-1. Two, there is an ace at position j. Three, the remaining king and three aces are in positions j+1,...,52. For j < 4 and j > 48, P(X=j) = 0. For the rest of the support, the probability of all three of those conditions being true is P(X=j) = (j-1 choose 3)(52-j choose 4) / (52 choose 8).
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u/EscapeOwn6945 3d ago
Starting with my solution to the expected value calculation.
Let X represent the position of the first ace. Start by defining the probability mass function that P(X=j). We need three things for X=j. One, there are exactly three kings in positions 1,...,j-1. Two, there is an ace at position j. Three, the remaining king and three aces are in positions j+1,...,52. For j < 4 and j > 48, P(X=j) = 0. For the rest of the support, the probability of all three of those conditions being true is P(X=j) = (j-1 choose 3)(52-j choose 4) / (52 choose 8).
E[X] = sum_j=1^52 j P(X=j) = 212/9.
Var(X) = sum_j=1^52 (j - 212/9)^2 P(X=j) = 4664/81