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));
>
>