Sunday, 18 June, 2006

Last night, a friend of mine said the following as we left the Casino:

"I know each spin is random, but if you'd followed the board you'd have made some money."

I replied, "There are no system of bets on Roulette that will consistently make you money. The House always has an edge against you and that applies on every spin of the wheel."

I just want to use this post to dispense with the mythology behind such systems. I wrote a private e-mail on this very topic in February and much of the material in here is very similar in nature.

We can summarise the position of my friend more precisely as follows:

The Roulette Wheel doesn't go "RED, BLACK, RED, BLACK, RED, BLACK...." but is more likely to go "RED, BLACK, RED, RED, BLACK, RED, BLACK, BLACK" so by playing with the wheel we should be able to improve our odds.

On the face of it this seems a fairly sensible thing to say but it is wrong. The mistake is in the second assertion: but is more likely to go "RED, BLACK, RED, RED, BLACK, RED, BLACK, BLACK"

This sequence is not more likely than the "RED, BLACK, RED, BLACK" sequence. In fact, they have exactly the same odds of occurring. This is because a number is red equally as often as it is black. The probability of a number being red is exactly 18/37, as it the probability of it being black.

When two events are totally independent of each other, the probability of the first event happening and then the second one happening is the multiplication of the two probabilities. A little example can help illustrate this:

e.g. Chance of heads on a coin: 1/2. Chance of rolling a 6 on a dice: 1/6. Chance of rolling a 6 AND a heads on a coin: 1/2 x 1/6 = 8%

So if we take "RED, BLACK, RED, BLACK, RED, BLACK", the probability of this occurring is:

 ProbabilityOfRed x ProbabilityOfBlack x ProbabilityOfRed x ProbabilityOfBlack x ProbabilityOfRed x ProbabilityOfBlack 18 / 37 x 18 / 37 x 18 / 37 x 18 / 37 x 18 / 37 x 18 / 37 = 1.3%

So what we see here is that since the probability of RED and BLACK are the same, the probability of any given sequence of RED and BLACKs are the same. The RED, BLACK, RED, BLACK sequence looks less random because the majority of the possible combinations of REDs and BLACKs consist of small runs of RED and BLACK but there is only one sequence where RED follows BLACK indefinitely.

How does this destroy my friend's theory? Well, suppose we had a rule that says:

If the last two numbers were the same colour, bet that same colour. If the last two numbers were different, bet the colour opposite colour to the last win.

Let's consider each possibility in turn:

• RED then RED: The next bet is RED which has a probability of 18/37 of winning. Betting BLACK would have the same chance.
• RED then BLACK: The next bet is RED, which has a probability of 18/37 of winning. Betting BLACK would have the same chance.
• BLACK then RED: The next bet is BLACK which has a probability of 18/37 of winning. Betting RED would have the same chance.
• BLACK then BLACK: The next bet is BLACK which has a probability of 18/37 of winning. Betting RED would have the same chance.

This is just one example of a possible betting strategy but hopefully you will see that all strategies of this type are doomed to failure. Betting strategies such as this are very seductive. The mind wants to see patterns in randomness that aren't really there and casinos exploit this to make money. People are far more likely to risk their money when they're working to a "strategy". The belief in their strategy let's them suspend their rationalist interpretation of the probabilities.

Remember, the reason they have cards that allow you to mark down the results of previous spins of the wheel is not because they're particularly nice people. It's because if people believe they have a system to beat the casino, they'll be far more willing to risk their money. The casino wants to encourage this behaviour since they no that there are no such systems.

Simon.

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