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applied approximate arbitrary assume ball drawn Bernoulli's theorem bility black balls cards characteristic function compound probability consider continued fraction convergent corresponding definition determined distribution function divergence coefficient double integral E(Xi equal equation expression find the probability formula given Hence hypothesis independent trials independent variables integer integral interval Jacob Bernoulli Laplace Laplace's formula large numbers law of large lemma Liapounoff's limit theorem Markoff mathematical expectation mean mutually exclusive n games needle normally distributed number of games number of points number of successes obtain occur player polynomial positive number power series preceding Prob proba probability of drawing problem proof represents the probability required probability respectively result roots ruined sequence Solution standard deviation stochastic variables sufficiently suppose theorem of total theory of probability tickets total number Tshebysheff's urn contains values variables Xi whence white balls