Math Puzzle Answer

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Game of Craps - Answer

The number of rolls of the dice you can expect in a game of craps is 3.376. Here's the solution:

The frequency (F) of each number (N) rolled is F(N) = (6 - lN - 7l) / 36.

Of all 36 possibilities, the 12 which have an N of 2, 3, 7, 11, or 12 end on the first roll.

For other values or N, the probability of ending on the next roll is P(N) = F(N) + F(7). So the expected number of rolls after the first roll for these values of N is 1 / P(N).

For all possibilities, the expected number of rolls is
E = 1 + ∑(F(N)/P(N)) = 1 + 2(3/9 + 4/10 + 5/11) = 557/165 = 3.376.