Beating the House: RouletteCameron LeslieKing Williams College001345-0046

Introduction

No one can possibly win at roulette unless he steals money from the table while the croupier isnt looking, is a quote by Albert Einstein that accurately reflects most opinions on roulette. Roulette is a casino game, which most people believe is impossible to get an edge on. The house edge is the percentage chance that the house or casino will beat a player. There are methods to get an advantage over the casino though, such as relying on a biased wheel or using a computer to track the ball. One suggested method to beat roulette is by spreading bets properly. Through this investigation, I want to see whether the methods that can be used to beat Roulette are possible. I chose this topic as I find it interesting and I am curious as to how you can beat a game, which is known by most professional gamblers for the fact that you cant get an edge on it. I wanted to see the different ways mathematics could be applied to beat the casino's roulette wheel. My interest in casino games has always been there as I have family friends who are professional black jack players who are able to beat the house. My interest in roulette arose specifically after I saw a video on online roulette, which contained a fool-proof way to beat roulette. The method used is known as the Martingale Strategy and is inefficient and ineffective. It would also require an enormous bankroll to be effective. It wouldnt work in a real casino but it did get me curious as to whether it was possible to beat a game as unpredictable as roulette.

RouletteThere are multiple types of Roulette, but for the sake of this investigation, it will focus on French or single zero roulette. American Roulette has an extra number, which is 00.[footnoteRef:1] This gives an even greater edge to the casino as opposed to the probability of winning being out of 37 it is out of 38. This investigation will be using the French version of roulette without the 00 number. European and French roulette have the same amount of numbers but a different table layout. Figure 1 shows the layout of a French roulette table.[footnoteRef:2] [1: The Wizard of Odds. "Roulette." Last modified March 2, 2013. wizardofodds.com/games/roulette/.] [2: Shelley, Ron. "French Layout-Single Zero Wheel."Wikipedia. 1986. Accessed March9,2014. http://en.wikipedia.org/wiki/File:French_Layout-Single_Zero_Wheel.jpg.]

The GameIn the game of Roulette, players choose to place their bets on either a single number or a range of numbers, the colours red or black, or whether a number is odd or even. To determine the winning number or colour, the croupier spins a wheel in one direction and spins a ball in the opposite direction on a small track around the circumference of the wheel. When the ball loses momentum, it falls into the wheel and lands in one of 37 (38 in American roulette) coloured and numbered pockets on the wheel. Figure 1

The types of bets in Roulette are important to understand, as each one gives the player a different probability of winning. This is due to different bets covering a different number of squares. Different bets can also have different payouts, which depend on the number of squares being bet on. There are two types of bets, an inside bet and an outside bet. An inside bet has a lower probability of winning and a larger payout. [footnoteRef:3] [3: The Wizard of Odds. "Roulette." Last modified March 2, 2013. wizardofodds.com/games/roulette/.]

Inside Bets Straight A bet on a single number or square Split A bet on two adjoining numbers (vertical or horizontal on the betting table) Street A bet on three numbers in a single horizontal line. Corner A bet on 4 numbers in a square. Six line A bet placed on two adjoining streets. Trio A bet on 0, 1 and 2 or 0, 2 and 3. Basket A bet on 0, 1, 2 and 3.Outside Bets Manque A bet on the numbers 1-18. Passe A bet on the number 19-36. Rouge ou Noir A bet on which colour the wheel will show (red or black) Pair ou Impair A bet on either an even number or an odd number not including zero. Dozen Bets A bet on either the first, second or third group of twelve numbers (1-12, 13-24, 25-36) Column Bets A bet on twelve numbers of one of the three vertical lines. (1-34, 2-35, 3-36)All of these different bets payout differently. The formula for calculating the payout that a bet will have is:

Where n is the number of squares the player is betting on. [footnoteRef:4] [4: The Wizard of Odds. "Roulette." Last modified March 2, 2013. wizardofodds.com/games/roulette/.]

Type Of BetNumber of SquaresPayout

Straight135

Split217

Trio and Street311

Basket and Corner48

Six Line65

Dozen and Column Bets122

Manque, passe, rouge ou noir and pair ou impair181

The payout is designed in this way so that regardless of the number of squares bet on, the casino will always get the same edge or expected value. The House EdgeThe house edge or expected is the percentage of the bet that you will lose on average in the long run. Calculating the expected value requires knowledge of how much money that will either be won or lost, the probability that the casino wins and the probability that the casino loses. Due to the fact that the house edge is how much the casino will win in the long run, it is not simply the probability that the casino will beat you.[footnoteRef:5] [5: PlayRoulette Online | Online Roulette Casino Reviews. "Roulette House Edge | Online Roulette Games House Edge." Accessed March9,2014. http://www.roulettestar.com/house-edge.php.]

The house edge can be calculated using expected value though, as the house edge is a long-run average. To find the expected value, let be the probability of an event happening. This means that to find the house edge we need to find the mean of . Assuming that a manque bet is made with 1, then there are two events that could happen. There could be a gain of 1 by the bettor or the loss of 1 by the bettor. If we let x be either +1 or -1 (win or lose) then the respective probabilities for are or . This can be more clearly shown in the following probability distribution:

+1-1

Considering that the expected value is the average of what will either be won or lost in the long run, the expected value would be equal to the mean of . The mean of or the expected value of X can be found using the formula:

This is the same as:

This formula[footnoteRef:6] is the sum of all of the values multiplied by the corresponding values. [6: Buckle, Nigel, Iain Dunbar, and Fabio Cirrito. Mathematics Higher Level (Core): International Baccalaureate. [Victoria]: IBID Press, 2007(521-526). ]

This would mean in terms of roulette and the above probability distribution, the expected value can be calculated as follows:

This means that for that if you are betting on a manque bet you would on average lose 2.7% of what your betting to the casino and would not gain anything. This is the expected value and the house edge. Making different bets, which have different payouts, has similar results. For example if a 1 split bet is made, two squares are bet on. Using the payout formula:

This means that that 1 bet would win a player 17. The probability of that bet wining is and the probability of that bet losing is. The probability distribution of this bet is:+17-1

If the same formula for the mean of as shown above, the result is:

This demonstrates why the payout varies depending on the amount of numbers that are bet on. It is so the expected value stays the same regardless. This has the benefit of making it very hard to make the house edge smaller. There are strategies though, which are supposed to give the player a slight advantage.Betting MethodsThere are many betting methods, which are supposed to give the player an advantage over the casino, but unfortunately most of them are not successful. An example of this is the martingale strategy. This is where a player makes a bet on a set of 18 numbers and doubles their bet after a loss.[footnoteRef:7] This is an unfeasible method though as table limits will prevent the amount of times that a bet can be doubled and also there is a limit to the amount that players can afford. Also, if the doubled bet wins, most of what is won is simply making up what was lost. The profit will be only what the original bet would have won, making it an inefficient method. [7: Top Ten Las Vegas Tips, plus our How-to-Gamble guide. "Betting Systems -- Gambling systems explained." Accessed March9,2014. http://vegasclick.com/gambling/bettingsystems.html.]

The James Bond method is a roulette strategy developed by writer Ian Fleming. It involves spreading bets around to try and get an advantage over the casino. It requires making three bets simultaneously; a pass, a six line and a straight bet.[footnoteRef:8] This strategy involves spreading your bets so certain amounts are placed on each of these bets. [8: How To Win In A Casino | Roulette Strategy at MyCasinoStrategy. "Roulette James Bond Strategy." Accessed March9,2014. http://mycasinostrategy.com/roulette+james+bond+strategy/1/MlW-gRWfIpSPM9O3I1KzcNWrcdOjMhOfMhKvYZe7gRKjU5OLU9OHUlKLgdaPId.]

70% of the bet is placed on the passe and 5% of the bet is placed on the zero for the straight bet. 25% of the bet is placed on the six line bet. If the amount of money that a player has to bet is 200, 10 would go on the zero for the straight bet. 140 would be placed on the passe and 50 is placed on the straight bet. If any of these bets won, they would each have a different return. The returns are as follows:

BetFormulaPayoutPayout Including Losses

Straight (10)

350160

Six line (50)

250100

Passe (140)

14080

Although there is a small return 25 out of the 37 squares now have bets on them. This means that the chance that the casino wins is now reduced to or 32.4%. The chance a player has of earning money is 67.6%. The probability distribution of this method is shown as: +350+250+140-200

The expected value or the mean of X can be found using the same formula as shown on page 6 of this exploration. The result is:

This would mean on average if a player keeps making the same bets with 200 spread as described in the method would win on average 26.6 pence for every pound that is be or 53 per 200 that is bet. This is a large return for a game that is thought to be impossible to get an edge on, which is why this is such a popular strategy. This result is incorrect though, as it doesnt incorporate the fact that receiving return from one of the bets means that what was bet on other numbers is lost. This means that the true probability distribution table is as follows:+160+130+80-200

This means the actual expected value is:

This means that on average for each of the 200 bets made, the return will be a loss of in the long-term. In the long term the expected value would still be in favour of the casino. Even though this appears to be a good method due to the fact that the number of squares the bettor covers is more than the casino, it infact has the same expected value as any other bet made on Roulette. This makes it a method of betting, which would lead to loss in the long run. A GeneralisationIt can be proven that even with multiple bets, the result will always be a loss through a generalisation of the formula. Two bets are made that each cover n squares. Let these two bets be W1 and W2. As W1 and W2 are both bets, they will both be positive numbers. W1 will cover n1 squares and W2 will cover n2 squares. The bets cannot both win so the winning of one bet must lead to the loss of the other. The information necessary to calculate the expected value is demonstrated in the table below. BetFormulaPayoutPayout including LossChance of Winning

W1W1W1

W2W2W2

The probability of both bets losing would be .

The expected value can be calculated by using this information. Both the generalised probability distribution table and the expected value are shown below.

W1W2 (W1+ W2)

As the wager is W1 and W2 together, to get the expected value per pound, you must divide the expected value by (W1 + W2).

This proves that the expected value will always be even with spread bets.Predicting the wheelThere are other ways to beat a roulette wheel. One method is to establish if a wheel has any bias. This is a possible solution, but is not worth trying. Wheels with biases are very hard to find, as when an obvious bias is revealed, the wheel is replaced so the casino can get back its edge. It requires viewing a wheel hundreds of times and if it is established that a bias exists, when someone wins consistently, a casino is likely to investigate and change the wheel. Another method, which has been proven to be successful, uses the initial position, velocity and acceleration of the ball to ascertain the area that the ball will end up in.[footnoteRef:9] The American institute of physics, has recently released a paper[footnoteRef:10] on this subject.[footnoteRef:11] The key behind the method shown in the paper is separating the path of the ball into four stages. These stages are: when the ball has a fast enough momentum that it remains in the rim of the wheel, when the balls momentum drop enough that it leaves the rim of the wheel, when the ball is rotating freely around the wheel but not in the rim and when the ball starts to hit the deflectors. For an accurate prediction, the angle of the wheel also needs to be included in the calculation. One consideration that made calculations within this paper easier though, is the fact that the only difference in every spin is the time the ball spends in the rim. The velocity of the ball when it leaves the rim will be the exact same in every case as the balls are all of the same exact size and weight. This means that by finding the point at which the ball leaves the rim is how the final location of the ball will be found. Edward O. Thorpe has also released a paper on predicting the outcome of roulette although it is not as detailed as the previous one.[footnoteRef:12] [9: The Free Library. "How to win at roulette." Last modified October 20, 2012. http://www.thefreelibrary.com/How%20to%20win%20at%20roulette.-a0305780128. ] [10: Full paper can be found at: http://school.maths.uwa.edu.au/~small/pdf/Chaos22-6.pdf ] [11: Small, Michael, and Chi Kong Tse. "Predicting the Outcome of Roulette."Interdisciplinary22 (2012): Accessed March9,2014. http://scitation.aip.org/content/aip/journal/chaos/22/3/10.1063/1.4753920.] [12: Thorpe, Edward O. "The Invention of the First Wearable Computer." Columbia University. Last modified January 20, 2003. http://monet.cs.columbia.edu/courses/mobwear/resources/thorp-iswc98.pdf.]

ConclusionUnlike in Internet roulette, which uses sophisticated random number generators, it is very possible to predict the outcome of a physical roulette wheel by using the velocity and initial position when the ball leaves the rim of the wheel. It is not possible though, to beat Roulette through spreading bets, even though it appears to be possible. The expected value for French roulette will always be , regardless of spreading bets or how many squares a better covers. A game, which was considered unbeatable for so many years by mathematicians, is now beatable due to computers, laser scanners and applying physics to the model. Without the aid of computers though, the casino will always beat a player in roulette.

Bibliography

Buckle, Nigel, Iain Dunbar, and Fabio Cirrito. Mathematics Higher Level (Core): International Baccalaureate. [Victoria]: IBID Press, 2007(521-526).

How To Win In A Casino | Roulette Strategy at MyCasinoStrategy. "Roulette James Bond Strategy." Accessed March9,2014. http://mycasinostrategy.com/roulette+james+bond+strategy/1/MlW-gRWfIpSPM9O3I1KzcNWrcdOjMhOfMhKvYZe7gRKjU5OLU9OHUlKLgdaPId.

PlayRoulette Online | Online Roulette Casino Reviews. "Roulette House Edge | Online Roulette Games House Edge." Accessed March9,2014. http://www.roulettestar.com/house-edge.php.

Shelley, Ron. "French Layout-Single Zero Wheel."Wikipedia. 1986. Accessed March9,2014. http://en.wikipedia.org/wiki/File:French_Layout-Single_Zero_Wheel.jpg.

Small, Michael, and Chi Kong Tse. "Predicting the Outcome of Roulette."Interdisciplinary22 (2012): Accessed March9,2014. http://scitation.aip.org/content/aip/journal/chaos/22/3/10.1063/1.4753920.

The Free Library. "How to win at roulette." Last modified October 20, 2012. http://www.thefreelibrary.com/How%20to%20win%20at%20roulette.-a0305780128.

The Wizard of Odds. "Roulette." Last modified March 2, 2013. wizardofodds.com/games/roulette/.

Thorpe, Edward O. "The Invention of the First Wearable Computer." Columbia University. Last modified January 20, 2003. http://monet.cs.columbia.edu/courses/mobwear/resources/thorp-iswc98.pdf.

Top Ten Las Vegas Tips, plus our How-to-Gamble guide. "Betting Systems -- Gambling systems explained." Accessed March9,2014. http://vegasclick.com/gambling/bettingsystems.html.

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