Biased Roulette Wheels
February 13th, 2017 at 10:29:36 AM
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I recently watched a video on how Gonzalo Garcia-Pelayo and his family exploited biased roulette wheels in Europe and the United States. You will find the video at www.youtube.com/watch?v=jnueFebjveE.
He used a BASIC program on a PC to decide whether a particular wheel had sufficient bias to make it exploitable, but details were not given. The Wizard addressed a similar problem on his roulette page by answering the following:
Dear sir, I "clocked" an automated single-zero roulette game for 8672 games. My predetermined number came up an amazing 278 times. I chose the number because of the wear and tear of the pocket. How sure am I that this number has higher probability than 1/37?
The Wizard analyzed the given data and concluded that there was a very high probability that playing that particular number would be profitable. I performed a similar but slightly simpler analysis and reached the same conclusion.
The correspondent says he is observing a single zero wheel but inquires about the probability 1/37, which applies to a double zero wheel. Since the payoff for hitting a single number is 35, the break-even probability of a given number is 1/35 and this is true whether the wheel is single zero or double zero. I think what the correspondent wants to know is if he can confidently accept the proposition that his number will be profitable to play. To answer this I propose a binomial distribution with p = 1/35. Using 278 successes in 8672 tries, the Stattrek binomial probability calculator gives 0.029332 as the probability that the result was from a break-even number, and so we accept the alternative: that the number was better than break-even at the 3% level of significance, which is the chance of being wrong.
The next question is how to bet. Should one place small bets or should one bet the house limit each time and not worry about being wiped out early? Should you increase your bet after a win and reduce it after a string of losses? Should your goal be to play as long as possible or to bet as many chips as possible, or doesn't it matter?
Or should you use the Kelly criterion, which is supposed to give you the maximum return from a favorable betting proposition? Our best estimate of the probability of our number coming up is 278/8672 = 0.032057. With p = 0.032057 and b = 35 the Kelly fraction is (0.032057 x 36 - 1)/35 = 0.0044016, which is $5 for a $1,000 bankroll. You increase your bet when you win and decrease it when you lose. This applies only to games favorable to the player.
I hope that the video and this discussion will generate interest in finding and exploiting biased roulette wheels.
He used a BASIC program on a PC to decide whether a particular wheel had sufficient bias to make it exploitable, but details were not given. The Wizard addressed a similar problem on his roulette page by answering the following:
Dear sir, I "clocked" an automated single-zero roulette game for 8672 games. My predetermined number came up an amazing 278 times. I chose the number because of the wear and tear of the pocket. How sure am I that this number has higher probability than 1/37?
The Wizard analyzed the given data and concluded that there was a very high probability that playing that particular number would be profitable. I performed a similar but slightly simpler analysis and reached the same conclusion.
The correspondent says he is observing a single zero wheel but inquires about the probability 1/37, which applies to a double zero wheel. Since the payoff for hitting a single number is 35, the break-even probability of a given number is 1/35 and this is true whether the wheel is single zero or double zero. I think what the correspondent wants to know is if he can confidently accept the proposition that his number will be profitable to play. To answer this I propose a binomial distribution with p = 1/35. Using 278 successes in 8672 tries, the Stattrek binomial probability calculator gives 0.029332 as the probability that the result was from a break-even number, and so we accept the alternative: that the number was better than break-even at the 3% level of significance, which is the chance of being wrong.
The next question is how to bet. Should one place small bets or should one bet the house limit each time and not worry about being wiped out early? Should you increase your bet after a win and reduce it after a string of losses? Should your goal be to play as long as possible or to bet as many chips as possible, or doesn't it matter?
Or should you use the Kelly criterion, which is supposed to give you the maximum return from a favorable betting proposition? Our best estimate of the probability of our number coming up is 278/8672 = 0.032057. With p = 0.032057 and b = 35 the Kelly fraction is (0.032057 x 36 - 1)/35 = 0.0044016, which is $5 for a $1,000 bankroll. You increase your bet when you win and decrease it when you lose. This applies only to games favorable to the player.
I hope that the video and this discussion will generate interest in finding and exploiting biased roulette wheels.
