Recherches sur l'intégration des équations différentielles aux différences finies, et sur leur usage dans la théorie des hasards.

P.S. Laplace


Two players A and B, of whom the respective skills are in ratio of p to q , play together in a way that, on a number x of trials, there lacks n of them to player A, and consequently x-n to player B, in order to win; the question is to determine the respective probabilities of these two players.

Let y(n,x) be the probability of B winning. Laplace finds an explicit formula for this probability. Let r denote p/q. In this case we may write y(n,x) = sum(r^k*binomial(x-1,k),k = 0 .. n-1)/((r+... .

> restart:

> y:=unapply(Sum(r^k*binomial(x-1,k),k=0..n-1)/(r+1)^(x-1),(r,n,x));

y := proc (r, n, x) options operator, arrow; Sum(r^...

We should note that the respective probabilities of the players must sum to 1. Suppose, for example, the ratio of the strength of Player A to that of B is r=2. Suppose further that Player A lacks 3

> value(y(2,3,8));


> value(y(1/2,5,8));