recherch_xiv.html
recherch_xiv.mws

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

P.S. Laplace

**PROBLEM
XIV.**

*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*
*to player B, in order
to win; the question is to determine the respective probabilities of
these
two players.*

Let
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
.

`> `
**restart:**

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

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));**

`> `

`> `