Lotto Nonsense: The World is Asking to be Deceived

These systems and their inventors always remind me of snake oil peddlers depicted in the old westerns. Usually a bit long-in-the-tooth and down-at-the-heels, these glib, fast-talking swindlers palmed off utterly useless health elixirs and salves on unsuspecting farmers and homesteaders. Of the many strategic systems claiming to increase one's chances of winning the lottery, let's focus on debunking the `Balanced Numbers' system.

This system was conceived of by an American woman who has been promoting herself as America's most trustworthy lottery expert for more than 30 years, and who has a huge following of gullible believers. Her books are all best sellers and have earned her much more money than she ever would have gained by playing the lottery. The Balanced Numbers system is one of the jewels in her crown. She lures faithful followers by claiming that the system is based on the principle of the `bell curve', expropriating, in this way, the name of one of the greatest mathematicians of all time, Carl Friedrich Gauss.

How is the system used? It is brilliantly simple. To illustrate, let's take the lottery game 6/49, in which six different numbers are randomly chosen from the series of numbers 1 through 49. For this lottery, the `balanced numbers' method advises players to choose six different numbers such that their sum will fall between 117 and 183. This advice is based on the fact that the probability distribution of the sum of the six drawn numbers can be accurately approximated by a normal distribution with its characteristic bell-shaped curve. The expected value and the standard deviation of the approximating normal distribution are 150 and 32.8: if r distinct numbers are randomly drawn from the numbers 1 to s, the sum of the drawn numbers has r(s+1)/2 as expected value and the square root of r(s+1)(s-r)/12 as standard deviation. About 68\% of the probability mass the normal distribution with expected value m and standard deviation s lies between m-s and m+s . That's why players of the 6/49 lottery are advised to choose six numbers that add up to a sum between 117 and 183. This strategy purportedly increases the player's odds of winning a lottery prize. Pure poppycock! This advice completely neglects the fact that there are many more combinations of six numbers whose sum falls between 117 and 183 in the middle of the sum's distribution, than there are combinations of six numbers whose sum falls between, say, 21 and 87, on the far end of the distribution. In any case, if you play this lottery, you are betting on the six individual numbers that are going to be drawn, and not on what their sum will be. Lottery win-systems claiming to increase your chances of getting rich are utterly worthless. If you want to be sure of doubling your money, my advice is this: fold it in half!

Roulette betting systems

Now that we know our onions, as it were,  when it comes to the lottery, we can move on to the subject of win-systems for roulette.  Let me be very clear: they do not exist! No matter how you play the game of roulette, you cannot escape the fact that, in the long run, the casino has a house edge. To illustrate, let's look at European roulette, which uses a wheel containing 37 numbers from 0 through 36. Blaise Pascal -- the French philosopher and mathematician -- is the father of the roulette wheel. He was not trying to invent a casino game, rather he was looking to invent a perpetual motion machine. Roulette emerged in the early 19th century as a glorious attraction in the casinos of Europe after legalization of the game by Napoleon in 1806.  Apart from the house number 0, there are 18 red and 18 black numbers on the roulette wheel. The croupier releases the ball into the game by rolling it onto the wheel in the direction opposite to the one in which the wheel is traveling. The number assigned to the pocket where the ball comes to a stop is the winning number. Players may bet on either a single number or on a combination of k numbers for  k =1, 2, 3, 4, 6, 12 or 18. If the ball stops on a number in the player's chosen combination of  k  numbers -- there is a probability of  k/37  that this will happen -- then the player wins 36/k times the amount staked; otherwise, the player loses the stake. In particular, if the player bets on red or black (k=18), then the player wins two times the amount staked if the ball comes to rest on that color. The number 0 is always a win for the house in European roulette, and this gives the casino an advantage over the player. Indeed, the expected value of a casino win on every euro staked on a combination of k numbers is positive and  is equal to  1- (k/37)*(36/k)=1/37. In other words, the casino's house edge is (1/37)*100%=2.70% regardless of which combination of numbers the player bets on. In the long-run, you simply cannot beat this house edge, no matter what system you use. Nevertheless, loads of betting systems claim to be winners. But there is no winning combination to be made from individual stakes that are individual losers. This can be shown mathematically, but for most people, the results are more readily convincing when achieved by means of computer simulation.

A popular betting system for roulette is the Labouchere system. Using this system, you must decide beforehand how much you want to win, and after that you make a list of positive numbers whose sum add up to this amount. The list is updated after each bet. You bet on red each time. For each bet, you stake an amount equal to the sum of the first and last numbers on your list (if the list consists of just one number, then that number is the amount of your stake). If you win the amount staked, you cross off the amounts you used from your list. If you lose, you add the amount lost to the bottom of your list. You continue in this manner until your list is used up (target amount has been achieved) or until you have lost all of your money.

The Labouchere system is an exciting system, and there is no reason not to enjoy using it for its entertainment value, but here also, with regard to long-term play using this system to play European roulette, your average loss per staked euro inevitably comes out at, or near the 2.70 cent mark. Let's show this for a specific situation. Say your win-target is 250 euro and that you have a bankroll of 2500 euro. Off you go to the casino. You configure a starting list made up of the numbers: 50, 25, 75, 50, 25, 25. In the first round, then, you stake 50+25=75 euro. If you win, your list is narrowed down to  25, 75, 50, 25, and if you lose, the list is extended to 50, 25, 75, 50, 25, 25, 75. It is worth noting that at a given moment the sum of the first and last numbers on the list might be larger than the amount of money still in your possession at that time. For example, let's say that at a particular moment your first and last numbers are 50 and 125, respectively, whereas you only have 150 euro left to bet. In that case you would bet the entire sum of 150 euro. In each simulation run, you begin with a bankroll of 2500 euro, and play according to the Labouchere system until either your bankroll has increased by 250 euro, or you have gone broke. Executing 100 thousand simulation runs produces the result that the player, on average, loses money with an average loss of about 0.0274 euro per euro staked   (the average loss per round was about 62 euro and the average amount staked per round was about 2265 euro). This simulated value of 0.0274 confirms that, for the long run, you cannot beat the house edge of 2.7% for the casino. We can conclude, then, that there simply is no system for playing roulette that leads to a player's win over the long run. If we are talking short run play, it is possible for a player to win at roulette. To illustrate this: we get a simulated value of 0.8865 for the probability that a one-time execution of the Labouchere system will end in a win for the player of 250 euro. This large win-probability for a one-time execution does not contradict the fact that the system is a losing one when applied over a large number of executions.

No matter which roulette system you try, Labouchère, Big-Martingale or the Fibonacci system, for play extended over the long run you lose on average 2.7 cents for every euro staked when playing European roulette. The roulette ball has no memory, and this means that it is impossible to make winning combinations of bets when each individual bet is a losing proposition. It is true that the one system is more exciting than the other, but when it comes right down to it, over the long run, each system has an average loss of 2.7 cents for each euro staked. These systems only differ from one another in patterns of betting and in the way in which they reconfigure a player's losses and wins. Attempts to influence your average loss over the long run by using a betting system for roulette are just as fruitless as the bizarre attempt of a long-ago despot to influence the ratio of newborn boys to girls by ordering that, as soon as a woman bore a son, she was prohibited from bearing any more children.  In the long run the gambler's chances in roulette are the same as those of a lamb in the slaughterhouse.

 A non-mathematical but nonetheless convincing proof of the fact that a winning betting system does not exist for the game of roulette is evident from the fact that casinos have never shown any resistance to the use of any such a system at the roulette table. Going back in time, there are gamblers who have broken the bank by detecting biased roulette wheels or by using electronic equipment to predict the path of the ball. The first and most famous biased wheel attack was made in 1873 by the British mechanical engineer Joseph Jagger. He recruited a team of six clerks to record the outcomes of the six roulette wheels at the Monte Carlo casino in Monaco.   Jagger detected a biased roulette wheel showing up the nine numbers 7, 8, 9, 17, 18, 19, 22, 28, and 29 far more often than a random wheel would suggest. In a cat-and-mouse game with the casino, Jagger and his team ultimately won two million francs, which was a fortune for 1873!

More recently, gamblers smuggled a laser scanner linked to a microcomputer in a mobile phone into the casino at the Ritz hotel in London, and using these high-tech devices they won 1.3 million British pounds on the evening of March 16, 2004. The scanner noted where the ball had dropped and measured the declining speed of the wheel. These factors were beamed to the microcomputer, which calculated the most likely section of six numbers upon which the ball would finally settle. This information was flashed onto the mobile phone just before the wheel made its third spin, by which time all bets must be placed. Then, the gamblers placed their bets in the area the computer had pinpointed as the ball's most likely resting place.  You only have to increase your odds by 3\% to go from losing on average to winning on average. The gamblers were arrested by Scotland Yard afterwards, but there were no legal grounds for charging them with `cheating'. They did not break the law by using a laser scanner -- the scanner has, after all, no effect on the outcome of the roulette wheel -- and they were permitted to keep their winnings. Of course, they were barred from ever again entering a casino. Casinos have the right to bar you -- without cause -- but especially if they notice you consistently beating the odds. At the end of the day, the only sure way to get rich by means of roulette is to open your own casino!

Author

Henk Tijms is honorary fellow of Tinbergen Institute and emeritus professor of operations research at Vrije Universiteit Amsterdam. He is author of various textbooks on operations research and probability theory.  This column is based on Chapter 9 of his book “Surprises in Probability—Seventeen Short Stories”, Chapman and Hall.