Figure 1: (a) Probability distributions P(k) calculated for different values of the spread parameter δ. (b) Lowest-to-highest probability ratio, ξ, as a function of δ. (c,d) Monte-Carlo simulation of Ω as a function of δ (c) and ξ (d) calculated for different values of N, i.e. the length of the sequence of the “last” numbers we bet on.

Figure 2: The Monte-Carlo simulation of the expected return Ω of a non-ideal roulette versus (a) the length of the sequence of the ”last” numbers N used by the gambler to place bets, and the lowest-to-highest probability ratio, ξ (b) shows the same as a 2D plot for the convenience. (c) Monte-Carlo simulation of the critical ratio of probabilities of realsations of the most probable and least probable number, ξc, as a function of N

Figure 3: Monte-Carlo simulation of the expected return Ω versus the length of the sequence of the “last” numbers N for different values of δ and lowest-to-highest probability ratio, ξ

Figure 4: (a) Monte-Carlo simulation of the expected return Ω as a function of the length of the row of “last” numbers N used by a gambler when placing bets and of the probability ratio ξ. The gray plane indicates Ω = 0. The calculation has been done using a linear probability distribution P(k) given by Eq. (9). (b) shows the functions P(k) taken at different values of the spread parameter β. (c) The critical value of highest-to-lowest probability ratio, ξc, versus N

Figure 5: The critical starting capital C (a), the corresponding average number of spins M between two catastrophic fluctuations (b), and number of successive unsuccessful spins S needed to spend the critical capital (c) as functions of Ω calculated for the different values of the average number of bets per spin . This calculation has been done with the probability distribution function (1).