Bunching of numbers in a non-ideal roulette: the key to winning\n strategies

Chances of a gambler are always lower than chances of a casino in the case of\nan ideal, mathematically perfect roulette, if the capital of the gambler is\nlimited and the minimum and maximum allowed bets are limited by the casino.\nHowever, a realistic roulette is not ideal: the probabilities of realisation of\ndifferent numbers slightly deviate. Describing this deviation by a statistical\ndistribution with a width {\\delta} we find a critical {\\delta} that equalizes\nchances of gambler and casino in the case of a simple strategy of the game: the\ngambler always puts equal bets to the last N numbers. For up-critical {\\delta}\nthe expected return of the roulette becomes positive. We show that the dramatic\nincrease of gambler's chances is a manifestation of bunching of numbers in a\nnon-ideal roulette. We also estimate the critical starting capital needed to\nensure the low risk game for an indefinite time.\n