How not to make money at a casino — or watch that denominator!


First, be assured, there is a sure-fire method for making money at a casino, but you’ll have to wait until (or skip ahead to) the end of the blog.

The associated problem of how not to make money at roulette or any other gamble that depend solely on chance came up in the March 7 edition of the SGU 5X5 podcast, in which the five skeptics of the Skeptics’ Guide to the Universe launch a coordinated five-minute attack on pseudo-science, superstition, alternative medicine and other forms of woo.

The episode in question dealt with a set of logical fallacies that live under the clumsy rubric of “the representative heuristic”.

Each skeptic gave an example or two, but the one that stuck with me was Jay Novella’s description of the gambler’s fallacy, which says that random events have a memory. For example, said Jay, roulette tables in Las Vegas sport a display of the last 20 or so rolls. The idea being — if you see that red has come up 15 out of the last 20 rolls, the chance of a black on the next roll is better than average.

A simpler example. Suppose I toss a coin five times and get five heads (there’s a 1 out of 32 chance of that happening, incidentally). What’s the chance of getting tails on the next toss?

The naive gambler reasons thus.

“We know in the long run that the proportion of tails is equal to the proportion of heads ( a half each ), so in the subsequent tosses, tails has to catch up.So a string of heads usually precedes an overabundance of tails.”

Wrong. On two counts.

First, in the long run, the proportion of heads never exactly equals to the proportion of tails. Lacking the stamina to toss a coin thousands of times, I set my computer the task of simulating a one-million coin toss. Five times. Here are the results.

         H      T
 samp1 499774 500226
 samp2 499413 500587
 samp3 499840 500160
 samp4 500666 499334
 samp5 500309 499691

In the long run, you would expect heads and tails to each come up 500,000 times. They never did. In sample 1 (samp1), tails won (452 times more frequently than heads). In sample 4 and 5, heads won. The matches are never perfect.

Second, strings of  10 heads are followed evenly in these trials by a head or a tail. I won’t bore you with the statistics.

So the naive gambler is wrong.

So how come the percentage of heads and the percentage of tails are so close together in the long run? And they are. As the following table shows, as percentages, the counts of H and T hover pretty closely over the 50 per cent mark.

        H    T
 samp1 50.0 50.0
 samp2 49.9 50.1
 samp3 50.0 50.0
 samp4 50.1 49.9
 samp5 50.0 50.0

This hovering gets more precise in even larger samples. I did one of 100 million.

      H        T
 49995819 50004181

As percentages —

   H      T
 49.996 50.004

Now heads comes in several thousand tosses over the expected 50 million mark. But as a proportion or a percentage, both heads and tails close in on the 50 per cent mark. It takes three decimal places to detect the difference.

That’s because the denominator in the second experiment is so large. The raw deviations from the expected 50 million heads are  buried by the huge denominator.

The reason that the percentage of heads is close to the percentage of tails in a large run is NOT that the coin remembers what’s gone before. It’s the size of the denominator.

The skeptics gave some other examples of this representative heuristic fallacy; most of them are “denominator” problems. Listen and learn.

Now — the only way to win at a casino. Remember the odds are stacked against the gambler and — gambling being a zero-sum game — in favour of the casino. So, to winning at a casino is obvious — own the casino.

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About aharmlessdrudge

Way back during the late Bronze age -- actually it was the 1950s -- all of us in high school had to take a vocational test to determine our interests and, supposedly, our future careers. I cannot remember the outcome, but I do recall one question that gave me pause. "If you were to win a Nobel prize, would it be in literature or in physics?" I hesitated over the question: although I enjoyed mathematics and science more than English class, I did have a couple of unfinished (and very bad) novels hidden away at home. I cannot remember what I chose back then, but the dilemma followed me to university, where I switched from mathematics to English and -- after a five-year stint in journalism -- back to mathematics. I recently retired as a professor of statistics. Retirement. What a good chance to revive my literary ambitions. I have finished a novel -- more about that in good time -- and a rubble of drafts of articles about mathematics and statistics is taking up space on my hard disk.
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