# Simulating Roulette with a Doubling System

Once, after I presented the roulette simulation in class, a student asked:
"My boyfriend has a system for beating roulette! He bets \$1 the first time. If he wins, he is ahead by \$1. If he loses, he doubles his bet, to \$2. Now if he wins, he gets back the dollar he lost plus one! If he loses again, he doubles his bet again. Each time he loses, he doubles his bet. Sooner or later, he has to win, so he always winds up ahead!
As Dave Barry would say, "I am not making this up." Of course, it's always "my friend" or "my spouse" who has the dumb idea, never "I". This betting system is actually quite old. In the 1750's, Giovanni Casanova used it, according to his memoirs. Evidently, it is not obvious what is wrong with this system.

Mathematically, you can say:

1. Each bet at roulette has an expectation of -2/38 times the amount you bet.
2. Betting more money on some bets just increases the negative expected value of those bets.
3. Betting repeatedly adds up the negative expectations. There is no way a bunch of negative numbers can add up to a positive number, no matter how many you add.
4. When you bet using a system that increases the size of your bets, you make it more likely that you will lose all your money sooner. That goes for this system and all its variants.
Yet the "boyfriend" is also right in saying that sooner or later you are bound to win, and each time you win you make up all your losses and wind up ahead.

What gives?

The answer is that the doubling strategy will, sooner or later, require you to come up with more money than you want to lose. At that point, you will have to abandon your system and take your losses.

This roulette simulator shows how that works. Typically, as it runs, you build up some winnings. Then will come a crisis: You will have lost enough times in a row so that your next bet will require more than the amount of money you planned to risk. If you choose to go ahead, you may get lucky and win, or you may get unlucky and lose. When you do lose, the simulation will tell you how many times you played, to show you how long you lasted.

In normal betting, \$1 at a time without a system, you can expect to last 190 plays. (190 is -10 divided by -2/38.) Let's see how long \$10 lasts for you using the doubling system.

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Betting \$1 at a time, \$1000 should be expected to last 19,000 plays.

The general argument that you cannot devise a system to beat roulette is based on the idea that each spin of the roulette wheel is independent of the other spins. It is possible to devise a system for some card games, such as Blackjack, because the cards dealt are not independent. Depending on what cards have already appeared, the probability of beating the dealer can change. Systems in Blackjack involve keeping track of the cards dealt and then increasing the amount bet when the undealt cards are such that the expected value of the game becomes positive.

For more on how doubling-the-bet-until-you-are-wiped-out works, See http://www.toxicdrums.com/doubling-system.html. He has an application you can download that makes a neat graph, showing the pattern of winnings over time!