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

Chances of a gambler are always lower than chances of a casino in the case of an ideal, mathematically perfect roulette, if the capital of the gambler is limited and the minimum and maximum allowed bets are limited by the casino. However, a realistic roulette is not ideal: the probabilities of realisation of different numbers slightly deviate. Describing this deviation by a statistical distribution with a width δ we find a critical δ that equalizes chances of gambler and casino in the case of a simple strategy of the game: the gambler always puts equal bets to the last N numbers. For up-critical δ the expected return of the roulette becomes positive. We show that the dramatic increase of gambler's chances is a manifestation of bunching of numbers in a non-ideal roulette. We also estimate the critical starting capital needed to ensure the low risk game for an indefinite time.