Classic Problems of Probability

Problem 9

De Moivre's Problem (1730)

Problem. A fair die is thrown n independent times. Find the probability of obtaining a sum equal to t, where t is a natural number.

Solution. Let be the score on the die on the jth (j = 1, 2, . . . , n) throw and let be the sum of the scores obtained. Then

where, for all j = 1, 2, . . . , n,

We now take the probability generating function of , remembering that the X_j's are independent and have the same distribution:

Now

Therefore,

where we have made use of the fact that . Now, is the coefficient of in and can be obtained by multiplying the coefficient ...