We can quote the bible too

I found this image while surfing the net. This of course is a paraphrasing of the actual indicated passage but the meaning is accurately captured.

Examining a Losing Roulette Strategy

In the casino, the cardinal rule is to keep them playing and to keep them coming back. The longer they play, the more they lose, and in the end, we get it all. Ace Rothstein (Robert De Niro), Casino

Blackjack is the only fair game in the casino. It’s the only game where you don’t expect a net loss after repeated play. Ever other game will cost you money in the long run. Take roulette for example. For any bet the expected earnings is negative. If you make the same bet over and over and add up what you have at the end you will always find that you have lost money. Therefore, if the casino keeps you playing, they will eventually win your money. Its statistically inevitable.

I came across a “strategy” to win at roulette that I wanted to discuss. A quick google search will reveal that this strategy is everywhere and some pages offer some discussion as to why it is supposed to work. The strategy is as follows:

Start by betting one dollar on red. If you win you get a dollar. If you lose you are down one.

If you lost the first round bet 2 dollars on red. If you win, you will be up a dollar (2 won minus the 1 you lost in the last round).

If you lost a second time bet 4 or red. If you win, you are up a dollar (4 – 2 – 1).

Repeat this process until you win once, thus being up a dollar. Every time you lose, you bet double the amount on red again. Eventually you will win and be up a net win of a dollar.

Now, before any of you go running to an online casino to win big money consider yourself warned: This strategy does not work. The flawed reasoning behind the strategy is simple. A progressive series of losing numbers is increasingly unlikely and by continually betting on the red you are essentially betting on the change of such a series occurring. Since that chance is increasingly small your probability of winning the next round in the series approaches one.

The problem with this logic is two fold. The most serious of these errors is that the roulette wheel has no memory of what occurred in the previous round. A bet on red has a 18/38 = 47.37% of winning regardless of what occurred in the past. There is a probability that is approaching zero but it’s the probability that a specific sequence occurs in a particular order. The sequence “black, black, black, black, black, red” is just as likely as the sequence “black, black, black, black, black, black” which is also just as likely as the sequence “black, red, black, red, black, black” or any other specific sequence of six outcomes. (Here i have neglected the difference between black/red and loss/red, which produces slight difference in the probabilities but does not change the discussion appreciably.) However, the probability of any of these three is approaching zero as the length of the sequence grows. What drives the probability to zero is the fact that the number of possible sequences is growing exponentially.

The second flaw in the logic is that the strategy ignores the fact that you have to have enough money to cover the geometrically increasing bet size. Everyone enters the casino with a finite amount of money to bet. Even if they allow themselves to mortgage their lives there is still a finite amount of money from which to draw. Since the probability of 5 loses and a win is a little less likely as 6 loses, the odds are, that next bet is going to cost you a lot. If you continue to play you will eventually hit a sequence where you cannot afford to bet the increased amount and collect your net dollar. And what is worse is that that final bet will wipe you out financially.

To make this more concrete I decided to model the strategy numerically by simulating repeated plays of a game of roulette. Starting with some initial pot size, the program will play the outlined strategy above. However, once the pot size hits zero or less, the pot size is set to zero for the remainder of the simulation – once you have lost all your money it stays lost. I then repeat this process thousands of time and average the size of the pot after n bets to obtain the expected pot size at the nth bet. The results are plotted below as a percent of the initial pot size.

The red, green, blue and black lines correspond to initial pot sizes of 100, 200, 300 and 400 dollars.  The results are striking and the strategy clearly fails.  Even for the large pot sizes the player is broke, on average, before 5000 pays have been made.  Furthermore, the player expects to begin losing money immediately.  This is steaming from the fact that a red bet has a negative expectation value and geometrically increasing your bet to cover your loses doesn’t change this fact.

The reason the larger pot sizes result in a slower rate of loss is because the initial bet of one dollar was used for each of the cases.  As a result, the player is betting a smaller percent of the pot per turn and thus the rate of loss is slowed.  If you scale the initial dollar bet accordingly all of these curves would fall on one another, as shown below.