The Gambler's Fallacy

The Gambler's Fallacy is the belief that a random event is more (or less) likely to occur because it has (or has not) happened recently. There are two different variations on the Gambler's Fallacy. Let's examine each of them.

“The deck has been cold to me lately, so I'm about due for some really great hands.”
“Bob has been a roll lately; I can't wait until his luck runs out.”

The Gambler's Fallacy is based on the principle that if we gather a large number of samples, the average of the samples will converge to the true average. In statistical theory, this is known as the Law of Large Numbers.

For a deck of cards, in the long run we will get our fair share of good hands and bad hands. People often reason that if they see an unusual number of good or bad hands, then they are “due” for more hands of the opposite type so that they will “even out”. While intuitively appealing, this isn't what the Law of Large Numbers says.

To simplify the mathematics, let's switch to coin tossing. Assuming a fair coin, the Law of Large Numbers says that if we toss the coin many times, we will see heads approximately 50% of the time. What happens if we flip the coin 10 times and only see heads 4 times (40%)? One might reason that to fulfill the Law, we must be more likely to see heads the next 10 times we flip the coin.

However, this can't be true if the coin is fair; it must always have a 50% probability of flipping heads. How do we resolve this conundrum?

The problem is that the Law of Large Numbers really does need LARGE numbers. Given the example above, the Law doesn't tell us anything about the next 10 coin flips. The Law tells us that if we flip the coin 990 more times, then we'll tend to be closer to 50%. Let's say that the next 990 samples are evenly split between heads (495 times) and tails (495 times). Overall then, we've seen heads 495+4 times, or 49.9%. That's much closer to 50% than 40%, without the coin “making up” for the initial shortage of heads.

“I'm running hot. I don't want to quit now while I'm getting all these great cards.”
“I'm having rotten luck. I should quit now and play again some other time.”

This version of the Gambler's Fallacy has the opposite meaning: it assumes that because an event has occurred a few times recently, it is more likely to continue. Returning to the coin analogy, a fair coin, by definition, always has a 50% chance of landing heads no matter how many times it has landed heads recently.

Of course, if the coin lands heads many times in a row, we might start to get suspicious that coin is not, in fact, fair, which perhaps is the source of this version of the Gambler's Fallacy.

Of course, in poker, there may be other reasons for having a streak of winning or losing, such as playing against better or worse players, which can be good reasons to quit or keep playing.

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