- 1 - STATISTICAL ROULETTE ANALYZER (SRA) Essential calculation

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1 - 1 - STATISTICAL ROULETTE ANALYZER (SRA) Essential calculation Contents: Introduction - substantiation 1.Exploiting the Dirichlet distribution. 2.Testing the roulette wheel. 3.Basic statistical parameters for a game on a biased wheel. 4.Standard deviation of the game`s basic statistical parameters on a biased wheel. 5.Utility program. Introduction substantiation Roulette a gambling device known for centuries. Currently it is present practically in every casino in the world and on the Internet. There are two kinds of roulette European (37 pockets, 0 to 36) and American (38 pockets, an additional 00 pocket). Assuming that the wheels are perfect and unbiased, the casino s advantage is 1/37 in the former case, and 2/38 in the latter. In recent years, owning the improvements made in the fields of optoelectronics and the information technology, new appropriately programmed devices have emerged which allow to predict the area of the ball s fall on the wheel thanks to the information on the mutual synchronization of the ball s movement and the wheel s momentum. This information is entered to the device during the initial stage of the spin. Such devices are concealed under clothes and are operated discreetly. This is, of course, illegal. It can be done during the classic procedure of the game, which is as follows: after the payout from the previous spin, the croupier encourages the players to bet, saying place your bets and spins the wheel. The ball and the wheel are in motion, placing bets is still possible, and only in the final phase of the spin the croupier closes the betting time, saying no more bets. The time of the spin, when betting is still possible, is used by the aforementioned devices to calculate the area of the ball s fall and to bet accordingly. The casino s effective defense against such procedure is the possibility to place bets only until the moment of the spin; the croupier says no more bets and makes the spin. This procedure is more and more popular, even now almost all automatic and online roulettes follow this rule, and in casinos this critical time is being shortened, and so: THE ONLY LEGAL WAY TO WIN ROULETTE IS PROPER A PRIORI KNOWLEDGE ABOUT THE WHEEL, THAT IS BEFORE THE SPIN, BEFORE THE GAME. Any roulette wheel has some faults which may be a result of a faulty construction or wear and tear. We are interested only in the faults which cause uneven distribution of the probability that the ball falls in specific roulette numbers. Thanks to the statistical analysis of the recorded results of the wheel s spins, this irregular distribution may be detected and the possibility to win may be assessed. It is the main goal of this work and of the utility program based on this (point 5).

2 1. Exploiting the Dirichlet distribution After some ordering (standardization) of the conditions during the spins ( help for the utility program, point 2), each spin may be treated as an independent event. This fact allows using the Dirichlet distribution to analyze the probability of winning specific numbers of the roulette. The data for this analysis are recorded results of the spins for specific roulette numbers. The Dirichlet distribution has the following form: 1> x1,..., xk>0 ; si> 0 ; i = 1... k x xk= 1 ; k >1 where: B (s) - the normalizing factor x i - probability density i of this event (hiting the number) k - number of possible events (roulette numbers) The Dirichlet distribution is a prior distribution in the Bayesian statistic. The f function specifies the probability density of the k group of the possible events the repeat number of which is n i = s i - 1 ; i = 1... k. In the case of roulette, the basic probability density distribution parameters for specific numbers have the following form: (2) ; ; ; i, j = 1...k ; i < j where : p i - average probability of hiting i this number wp i - average probability variation of hiting i this number cov ( x i, x j ) - covariance of the probability of two numbers (possible pairings) n i - number of recorded spins for i this roulette number n - sum of recorded spins for all roulette numbers k = 37 - European roulette k = 38 - American roulette Since in the further discussion what interests us are only the values p i > 1/36 and n > 200, it is possible, with a very reasonable approximation, to assume that the distribution of the probability density for the discussed roulette numbers is a normal distribution. 2.Testing the roulette wheel.. The main goal of the test is to check if it is possible to win on a tested roulette wheel on the confidence level equal to To this means it is necessary to design a bet which provides the optimal value of the potential total advantage. As it was mentioned above, we are interested only in the roulette numbers which are potentially won, i.e. those, of which p i > 1/36 (European and American roulette). The potential advantage on such number is: (3) Mi = 36 pi - 1> 0 ; pi = (2)

3 - 3 - Let s assume that there are m numbers. A problem emerges how to bet those numbers most efficiently, i.e. how to distribute the whole bet on specific numbers for testing of a wheel was effective. More about it on the next page. The betting is: (4) ; ; i = 1... m ; Mi = (3) Relevant for the European and the American roulette. The design of this bet uses the Kelly criterion. As it was mentioned, the probability density distribution for the considered roulette numbers is a normal distribution, and the total, potential advantage M is a linear function of the values of average probabilities (formulas 3 and 4), then the distribution of the density probability of this advantage is also a normal distribution. The standard deviation of the total, potential advantage M is: All values are respectively for the European and the American roulette (2)(4), and: - k = 37 European roulette - k = 38 American roulette The values i, j, select all possible pairings for the m numbers potentially required. The value of the potential total advantage M is specified with the dm error, but the zero level which the M value is related to, is also specified with some error. In order to really assess the potential advantage, we calculate the standard deviation for the same roulette numbers assuming M = 0 and the same distribution of the total bet of y i. Naturally, the total advantage, with such assumptions, equals 0, but the standard deviation equals: where: The values i, j, k - as above. By applying the properties of the normal distribution standard deviation, we indicate: (7) Z = dm0 ; dm0 = (6) In this discussion, the number specifies confidence level of 0.995, it is an accepted zero level for the potential advantage. The effective, potential advantage is therefore: (8) EM = M - Z ; M = (4) ; Z = (7)

4 - 4 - With the above assumed confidence level it is now possible to provide the condition which allows acknowledging that the advantage is not potential but real, i.e. the wheel is biased and it is possible to win with it. This condition has the following form: (9) ; dm = (5) ; EM = (8) It demonstrates that the common area of the normal distributions of the advantage with the average value equal to 0 and with the average value equal to M is less than This fact guarantees (at the confidence level of 0.995) that the value of the EM advantage is > 0, and so it is possible to win. Distribution of the total, potentially won bet on the potentially won numbers y i (4) gives always value (9) very close to the possible minimal value, so it allows for effective testing of a roulette wheel. In the (9) condition is met, i.e. the wheel is biased and it is possible to win on it, it is possible to calculate the basic statistical parameters of the game on this wheel. Assuming the zero level (7) results in making a correction to the average values of the probabilities of the won numbers p i. The effective average probability for those numbers assumes the following form: (10) ; i = 1... m ; z = (7); pi = (2) It should be noted that the values (10) will be applied to calculate the basic statistical parameters of the game on a biased wheel, but to calculate the standard deviations of those parameters, the values p i (2) will be applied (as it has been done for the EM values). 3.Basic statistical parameters for a game on a biased wheel. The presented proposition to be in the utility program (point 5) is named suggested bet. Naturally, it is the same as the optimal test bet (4). As (9) is met, it is possible to calculate the basic statistical parameters of the game. a) total advantage (11) EM = (8) b) total bet It is known how to distribute the betting on the won numbers (4), but its total value is still not known. The value of the total bet has the following form: (12) ; DK = (12b) where: EM - total advantage (8) BR - funds (bankroll), money for the game (12a) ; dm = (5) ; EM = (8)

5 - 5 - The kf factor changes from 0.3 to 0.5, depending on the dm/em values, and so from (9), the lower (9) is, the kf and the total bet is higher, but its maximum value reaches only half of the value resulting from the Kelly criterion. The kf factor is an attempt to include the influence of the atmospheric conditions on the wheel and the ball. ; pp i = (10) ; y i = (4) Formula (12) includes the Kelly criterion. The DK value is the original contribution of this work s author. Usually at this stage the variance value of the game is applied, which is only an approximation of the specific DK value. The DK value is an average win of any won bets in the units of the TB total betting. This value is the essence of the Kelly criterion denominator. c) betting on selected roulette numbers ( suggested betting ) (13) Bi = yi TB ; i = 1... m ; TB = (12) ; yi = (4) d) SCORE value (explained in help of the utility program point 3) (14) ; EM = (8); DK = (12b) e) risk of ruin (15) ; kf = (12a) f) win in 100 spins (16) W100 = 100 EM TB ; EM = (8) ; TB = (12) g) standard deviation for 1 spin (17) ; TB = (12) where: W - the game s variation is equal to: h) standard deviation for 100 spins (18) SD100 = 10 SD1 ; SD1 = (17) ; y i = (4) ; pp i =(10)

6 Standard deviation of the game`s basic statistical parameters on a biased wheel. These standard deviations are named "statistical error" in utility program (point 5) ("Gallery"). In the above expressions in which the Kelly criterion factor exists, directly or indirectly, is treated as a fixed factor which is not the source of the standard deviation. The same goes for the kf factor. a) standard deviation of the total advantage (19) dm = (5) b) standard deviation of the total betting (20) dtb = 0 c) standard deviation of betting on specific roulette numbers (21) dbi = 0 ; i = 1... m d) standard deviation of the SCORE value (22) ; dm = (5) ; EM = (8) ; DK = (12b) e) standard deviation of the risk of ruin (23) drr = 0 f) standard deviation of the win in 100 spins (24) dw100 = 100 TB dm ; dm = (5) ; TB = (12) g) standard deviation of the standard deviation for 1 spin where: TB = (12) ; wp i = (2) ; n - sum of recorded spins (25a) di = (36 yi- 1) 2-72 M yi - 1 ; M = (4); pi = (2); y i = (4)

7 - 7 - All values respectively for the American roulette (k=38) and the European roulette (k=37). The i, j values select all possible m pairings on the bet roulette numbers. h) standard deviation of the standard deviation for 100 spins (26) dsd100 = 10 dsd1 ; dsd1 = (25) Certainly, if sum of recorded spin n is going to infinity all standard deviations of basic statistical parameters (statistical errors) are going to Utility program. The calculations provided above constitute the most important part of the algorithm which constitutes the basis for the utility program. Apart from the fully presented above suggested betting, there are options in the program of two more betting options: simple bet and your bet. The essence of the simple bet (if it exists) is explained in the introduction of the help bookmark. Your bet requires a slightly deeper explanation. The suggested betting is a kind of a game proposition on a biased wheel, presented by the program s author, it is a safe proposition, i.e. with considerably low risk of ruin. Not every user of the program needs to agree with it, e.g. some prefer a more aggressive game, with higher betting. Your bet may serve also to adjust the betting to the rates on the wheel and the betting limits. It is worth to note that your bet option is available at any moment of the program use, and so it also plays the didactic role by showing what the game on an unknown wheel or illconsidered bet may lead to. The calculation of the basic statistical parameters and their standard deviations for the simple bet and your bet follows similar method as in case of the suggested betting. The utility program begins the analysis of the recorded spins if their sum exceeds 200 (n > 200). It is in accordance with the assumption made in the above calculations (point 1). To take into consideration the calculation it is possible to approximate results minimal value of total bet advantage EM of a biased roulette wheel as a function of number recorded spins n: for n = 500 minimal value EM = 0.18 n = 3000 EM = n = 6000 EM = 0.05 n = EM = 0.04 n = EM = It means - if for the number n recorded spins the wheel is not biased corresponding EM shows, approximately, maximum value of total bet advantage when the wheel will be biased for more number of recorded spins n. Author of the program advises (big experience) to break recording spins at n = 6000 if the wheel is not biased. A user can record and introduce any number spins n if he wants.