The Gambler’s Fallacy vs. Regression to the Mean

A croupier spins the ball of a roulette table. (Photo by Michele Spatari / AFP)

Longtime gamblers on everything from sportsbooks to casino games are probably familiar with the term “Gambler’s Fallacy.” Essentially, it’s a misreading of how odds—and things in the world—work.

Gamblers who have attempted to use analytics to help improve their success rate have probably heard the team “Regression to the Mean.” It’s basically a way to use the understanding of how odds work to better predict future performance.

The Gambler’s Fallacy basically involves the assumption that recent performance won’t continue because another outcome is “due.” And Regression to the Mean… well, it seems to say the same thing—that recent performance won’t continue.

So how can one be the cause of loss and lamentation since the beginning of time and the other be a new-school way to better game the system? The key is the subtle but important difference between the two. Let’s look at each in-depth to find it.

What Is the Gambler’s Fallacy?

Let’s go back to the oldest and simplest bet there is: Heads or tails. Flipping a coin and trying to predict the outcome obviously involves a 50/50 proposition, since there is an equal probability of the coin landing heads up and tails up.

Let’s say you’re betting on a coin toss repeatedly, and the coin comes up heads four times in a row.

That’s an unlikely thing to happen. In fact, the probability of that happening is one in 16—pretty rare, indeed.

As a gambler, you know that rare things don’t happen very often (hence the name), and a coin coming up heads FIVE times in a row is even MORE unlikely (one in 32 probability, actually). So clearly, the coin will probably come up tails on the next toss.

To put it simply, we’re due for a tails.

That one simple word—due—is the Gambler’s Fallacy in a nutshell. The fallacy is the belief that because something unexpected has happened more often than expected (in this case, heads coming up), it will happen less often in the immediate future.

The logic seems simple—the odds of heads and tails are each 50 percent. That’s not going to change. So if we’ve had lots of heads, we’re going to have lots of tails to balance it out.

That’s true … in the long run. But in the short run, literally, anything can happen. Despite the logic of the fallacy, the odds of tails coming up on the next toss is the same as it ever was—50/50.

“But what about the 1 in 32 chance of getting five in a row?” comes the counterargument. This part of the fallacy lies in conditional probability. Before we started tossing the coin, the odds of five straight heads was 1 in 32. Now that we’ve tossed four in a row already, the odds of getting a fifth? 1 in 2.

What Is Regression to the Mean?

The University of North Carolina basketball team has had an up-and-down season. At the end of December, they were coming off a 29-point loss to Kentucky and had previously been embarrassed in games against Purdue and Tennessee, and were 8-3 on the year. They were solid, but not great, from three-point range and solid, but not great, in defending the three.

Then came back-to-back games against Appalachian State and Boston College. The Tar Heels were unstoppable in those games, winning by 20 and 26 points. UNC shot 7-of-16 from three against App and 11-of-23 against BC, 87 percentage points higher than they’d shot all season. They allowed the two foes to make just 12 of 58 threes, which was 118 percentage points lower than they’d allowed up to that point in the season.

During any season, teams will have hot and cold stretches. Regression to the Mean says that the long-term performance is what to expect will come next. So, while the postgame quotes from the players and coach said that the team had “figured things out,” a savvy gambler understood that the performance wouldn’t continue.

Sure enough, the Heels have lost three of their eight games since. They’ve shot .357 from three, a little bit lower than they’d been shooting before the App State and BC games, and they’ve allowed opponents to shoot .371, about what they were shooting before that two-game stretch.

Simply put, Regression to the mean says to expect things to return to the established odds after an unlikely event. So, after four straight heads, we can expect about five heads over the next 10 tosses.

What’s the Difference?

Like we said, it’s subtle but important. The Gambler’s Fallacy says that when something happens more often than expected, it will happen less often in the future. To balance things out. That’s not true.

Regression to the Mean says that when something happens more often than expected, it will happen about as often as it usually does in the future.

UNC hit 15 of 27 threes in their last game, a blowout win over NC State. The players and coaches said all the same things afterward about being on the right track now.

What does that mean for their next game? The Gambler’s Fallacy would say we should expect them to hit far below their season average on three-pointers—maybe a .200 shooting percentage or worse. Regression to the Mean would say we should expect them to hit about their season average–.370 to .380.