Het probleem van de ruïnering der spelers

M. Meinesz

Statistica Neerlandica, 1956, vol. 10, issue 2, 87-97

Abstract: 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.

Date: 1956
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