In Poker and Philosophy: Pocket Rockets and Philosopher Kings, edited by Eric Bronson, and published by Open Court, there is an essay by Gregory Bassham and Marc Marchese titled “Don’t Play on Tilt! Avoiding Seven Costly Critical Thinking Errors in Poker“. As the title of the essay suggests, it is about seven types of critical thinking errors that bad or tilting poker players make too often and that any poker player needs to avoid making in order to play well. One of these critical thinking errors is falling prey to the fallacy known as “the gambler’s fallacy“.
The gambler’s fallacy is a mistake in reasoning when a person believes that the likelihood of some event can be affected by some other past independent events. For example, suppose I flip a fair coin and it lands on tails three times in a row. Suppose you think that on the next flip of the coin it will be likely to be heads, because it landed on tails three times in a row. Your reasoning would be fallacious. The likelihood that the coin will land on heads is actually 50-50, and the three prior flips of tails have no bearing on the future likelihood of any given flip. In short, the past doesn’t affect the future when it comes to coin flips. To think it does is to make the gambler’s fallacy.
Bassham and Marchese give some good examples to illustrate the gambler’s fallacy. But I take issue with the following example they give.
In basketball, imagine a player who is a career seventy-percent free throw shooter has gone 5 for 5 from the line so far. He goes to the line again. You might hear the TV commentator say he is due to miss this one, which implies that he is more likely to miss the next free throw rather than make it. Wrong! The odds say he will probably make it (seventy-percent chance he will make the shot).
Is it really a seventy-percent chance he will make the shot? I’m not so sure. There may be reason to believe that his five free throws are not independent. There may be a causal explanation for his unusually hot hand, and if there is, then the odds are not seventy-percent that he will make the next shot. Suppose the player had missed the first five free throws. Wouldn’t it be reasonable to suspect that something is wrong with the player, that he might be sick or nervous or have low morale or something else that is affecting is game on this particular night? My point is that the events in a given basketball game, unlike flipping a coin, are dependent, and thus the gambler’s fallacy doesn’t apply.
When I asked the math blogger who writes for Puzzling Thoughts about figuring out the odds of a 90% free throw shooter making a free throw after missing the first five in a row, he confirmed my view. Here is what he had to say in one of his comments.
This problem can be thought of in terms of conditional probability. If this was a real world problem, I would assign a conditional probability to the chance that a player makes a free throw given that he has missed the last x free throws based on empirical data, denoted P(player y makes a FT|player y missed x previous free throws). The formula for this probability is the probability of the intersection of the two events (which is not their product since they aren’t independent) over the probability of him missing the x previous FTs. This would be a difficult modeling task, however, because you may not have ever seen a given 90% FT shooter miss 6 in a row so there may not be any empirical data, in which case the probability would have to be extrapolated from the data you do have, and hopefully a statistical analysis of that data would show some sort of significance.
In poker one can make an error in reasoning that is the gambling fallacy, and one can also make an error in poker by confusing the gambler’s fallacy with the issue of conditional probability. Let me explain.
Bassham and Marchese give the following good example to illustrate the gambler’s fallacy in poker:
If you’re holding a pair after the flop, what are your odds of getting three of a kind on the turn or the river? Experienced poker players may know that the answer is less than ten percent. Knowing this statistic should be helpful in playing this game. However, knowing this percentage can also lead to committing the gambler’s fallacy. You’ve been playing all night with your buddies. For the last several hands you’ve been dealt a pair but so far you haven’t been able to land trips after the flop. On the next deal you get a pair of 10’s. You decide that you’re long overdue for the trips, so before the turn you bet high. You have just committed the gambler’s fallacy and most likely you won’t get the third 10 that you are desperately hoping for.
This example, as I said above, is a good illustration of the gambler’s fallacy. But I think it looks similar to the following example, which is not the gambler’s fallacy.
You’ve been playing hold’em all night with your buddies, and you’ve lost the last five hands where you’ve called with trips. Hence you figure the next time you call with trips you are due to win.
Is this good reasoning? Definitely not. Is this the gambler’s fallacy? No. Whether you win or lose a given hand depends on many factors, including what cards you’re holding. The reason that you’ve lost the last five hands with trips might be because some other player knows what you’re holding and knows that he has you beat. So if you’ve been losing all night at poker, you might think you’re due to win, because your losing streak can’t go on much longer. If your reason is based solely on what cards you’ve been dealt, then you’re committing the gambler’s fallacy. However, if your reason is based on other factors, then you are reasoning poorly, but you are not committing the gambler’s fallacy. And whatever fallacy it is, it can cost you a lot of money.