O'Reilly logo

Classic Problems of Probability by Prakash Gorroochurn

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

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

img

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

img

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

img

Now

img

Therefore,

img

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

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required