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Some Thoughts on Roulette Screens, from Victor Niederhoffer
All Casinos show a lighted screen indicating the colors and numbers (but not the sectors) of the last 18 or 20 spins on modern roulette machines. The idea is to encourage players who believe that the past spins are indicative of the future. The screens provide a nice exercise in probability theory, the influence of multiple comparisons, the decisions between patterns and randomness and the revisions of hypotheses based on subsequent events; all areas very close to proper decision making in our field.
I am just back from Las Vegas and do not wish to work out the numbers in my head, so a complete post on this subject will have to wait until I get out my binomial tables, however, the first question is: What is the probability of a unusual occurrence, say three of the same number -- a sticky wheel -- in 18 spins? Since it's 1/38 that a given number will appear, the chances of three or more of a given number is the binomial probability of three successes or more out of 18 with a probability of 1/38. That chance is 0.011. The second stage is to compute the probability that none of the 38 numbers will occur these three times or more. That would be (1-0.011) to the 38th. That gives is 0.65, thus the chance that there will be at least one number with the three or more hit rate will be 0.35. But of course there are four roulette machines at the average casino and a counter is likely to pass by the machines 10 times each to find the one that's not true. So that's 40 times to find a event that has a 0.35 probability. For that not to happen is a 1 - 0.65 to the 40. That is so close to zero that finding something is practically certain. Let's call it 95%.
Now assume you have found that that event happened. Let us compare the possibility that it occurs by randomness to the chance that the machine is biased and that there is a defect somewhere. Let us say if there is a defect that the chances are 1/20 that 14 numbers will occur and 1/100 that the other 24 will occur. With such a machine the chances of finding a run of three would be say 99% with the same searching pattern as before. Before you start you figure out that it is 90% that no machine will be defective and 10% that it will be defective.
Okay, you have observed a success. Which hypothesis is now more likely? The likelihood of the first hypothesis is 95/99ths as great as it was before, and the likelihood of the second hypothesis is now 99/95ths as great as before. That brings the first hypothesis down to 85% and the second one up to 15%. You can see that you would need about 8 separate instances of a given machine in independent trials giving 3 or more successes before you reached 50-50 chance that the machines were defective.
Similar reasoning is appropriate for using patterns to conclude that non-randomness occurs in the market, and that it will continue. The key is repeated instances of seemingly non-random behavior.
From the Chairman of the Old Speculators Association:
Reading your piece on Las Vegas and roulette wheels reminded me of a curious story. When I was 23 or so and working as an actuary at a large insurance company, I had a kid working for me as an actuarial student. I say kid, though he was probably a couple of years older than I. Anyway, he went to Las Vegas on vacation and returned to report on his activities at the roulette table. Being of an actuarial bent, he had watched the table for some while before starting to bet. During that period, he had kept track of what numbers came up. I naturally assumed he was looking for a bias in the table. No, as it turned out, he was looking to determine which numbers had not come up, because he figured they would be overdue. I kid you not. The punch line here is that he wound up as CEO and Chairman of the Board of the company. It's an interesting commentary on large insurance companies.