We humans aren’t wired to understand randomness.
For example, when two events happen in sequence, we have a tendency to believe that the first one caused the second.This is extremely helpful when it protects us from touching a hot burner a second time, or from poking around a bees’ nest. It’s not as helpful when it causes us to believe that blowing on the dice will help us to avoid rolling a seven.
Why we struggle with randomness
To a caveman, the cost of not understanding randomness was small. He might waste his time doing an unnecessary rain dance or two, but assigning causality where there was none wouldn’t kill him, the way failing to do so might.
As a result of evolution favoring the shortcuts that helped us survive, we evolved without much grasp of the concept of randomness. We’re now subject to several biases regarding random events, one of which is the subject of today’s post.
Are you guilty of this one?
One such bias is so prevalent, it has earned the name the Gambler’s Fallacy.
When a roulette wheel lands on black five spins in a row, we want to say that it’s “due” to hit red. Similar for a coin flip that comes up tails several consecutive times; lots of us feel it owes us a few reds to get back to even.
We know these inanimate objects have no memory, and that consecutive trials are independent. But we also believe in the so-called “Law of Averages,” which tells us that deviations from the theoretical mean should even out over time.
So which is correct?
A coin toss example
Let’s assume a coin is perfectly fair. It should come up heads half the time, tails the other half. Now let’s say we start flipping this coin and (amazingly) get 20 heads in the first 20 tosses.
My question to you is this: Given this history and the fact that the coin is fair, how many heads should we expect in the first 1000 flips?
A lot of people will answer “500, since over time the coin will be fair.” But that’s wrong.
When you say “500,” knowing that the first 20 flips were heads, you’re predicting that of the next 980 tosses, only 480 will be heads, while the other 500 flips will be tails.
Doesn’t sound very fair, does it?
The correct answer is that we expect 510 heads, 490 tails after 1000 tosses. Why? We start with the 20 heads. We have 980 flips to go, of which half (490) should be heads, while the remaining half should be tails. (Remember, the coin is assumed to be fair.) That leaves 510 heads and 490 tails.
Another way of looking at it: Heads has been given a 20-flip head start, we have a fair and memoryless coin, so after 1000 flips, we expect there to still be 20 more heads than tails.
But what about the Law of Averages?
Despite what you’ll hear almost nightly from a naive sports commentator somewhere, there’s no such thing as the “Law of Averages.”
What there is, however, is the Law of Large Numbers, which says that over time, the percentage by which the number of heads differs from half will go to zero. So what this law says about our coin flipping experiment is this: After a million, a billion, or a hundred trillion flips, we’d still expect heads to be winning by 20, given that it started out that way.
But in the context of such enormous numbers, 20 is a tiny percentage, and it becomes even less significant as the number of flips grows. So the number of flips that are heads converges to one half, even without “making up for” the run of 20 heads at the start.
In short: The Law of Large Numbers does not say that we should expect more tails than heads in future flips, only that the effect of the 20 heads will become negligible after many flips.
Watch and listen closely, and you’ll notice this fallacy all the time. The logic in the following examples is equivalent to assuming that if a roulette wheel has come up black a few times in a row, it’s bound to come up heads.
- You play poker and throw away every hand for half an hour because you’re being dealt trash. With every passing hand, it feels like a playable hand becomes more likely. (Of course it isn’t. Your chances of getting a playable hand now are exactly what they were before the run-o-trash started. Same goes for flush and straight draws: Lots of misses doesn’t make you due to hit one, so you shouldn’t adjust your play when you get frustrated.)
- An NFL team wins both games against a divisional opponent one season, then faces them again in the playoffs. Someone on ESPN tells us “how hard it is to win three games against a team in this league.” (Sure, but not once they’ve already won two of them. Two wins here does not make a loss “due.” Now it’s about winning one game against a team that’s likely not as good as them. If the reasoning is that the weaker team makes adjustments, fine, but I don’t think that’s what they’re saying.)
- An NHL team gets down 3-1 in a playoff series. They win the next two games. Now radio hosts tell us that they probably won’t win Game 7, because so few teams have ever come back from a 3-1 deficit. (Same idea here. Once they’ve tied it up 3-3, another win would make it three in a row. Three playoff wins in a row is unlikely, but once two of them are in the books, a team isn’t “due” for a loss any more than they were at the start.)
- Someone playing a slot machine for an hour with no luck gets up to hit the ATM for more money. When he does, he parks his wife in the seat so that nobody can get in on his machine, which is sure to start paying out after such a long drought. (By now, you get the point. The trials are obviously independent, so it doesn’t matter how long it’s been since the machine hit.)
The sports examples are excusable if there’s a good reason why the outcomes of the games shouldn’t be independent (psychology, shifting home field, adjustments by a team that has lost several games in a row). Barring that, though, I see them as gambler’s fallacy.
There’s a statistics concept known as reversion to the mean which, at first glance, seems to be inconsistent with independent trials, but in fact, it isn’t. That’s the subject of a future post, and if you like what you read here, I hope you’ll subscribe to get free updates. And please, let me know in the comments what you think.
Want more math? More basics like this? Harder stuff?
Let me know. Thanks for reading.