Het probleem van de ruïnering der spelers

The gambler's ruin.
When a single trial has two possible outcomes A and B, with probabilities p and q( p +q= 1), a succession of these trials forms a so-called Bernoulli chain. The well-known result for the probability of n times A and m times B is
In this article we consider the ruin problem, in which the initial capitals of the gamblers are a and b, respectively. In stead of a Bernoulli chain we then have a Markoff chain, with coefficients that are less simple than the ordinary binomial coefficients.
A more general expression (formula 1) is obtained for the probability distribution of the gambler's profit after a certain number of games, provided none of them became ruined beforehand. The probability for ruin after a certain number of games is a special case, similar to the results of Lagrange, Laplace and others, but appears in a form, more suitable for numerical calculations.
Some other results, obtained through the same method as developed in this paper are indicated.