The gambler's fallacy, explained

What the fallacy is

The gambler's fallacy is the mistaken idea that independent random events balance themselves out in the short term. After a run of reds, black feels overdue. After a string of low rolls, a six feels as if it must be coming. After a lottery number has not appeared for months, it looks like a good bet.

The events are called independent because each one has no connection to the last. A coin has no memory. It is the same lump of metal on the sixth flip as it was on the first, and it cannot know what it did before. Nothing has been used up and nothing is owed.

The night black came up 26 times

The fallacy is sometimes called the Monte Carlo fallacy, after an evening at the Monte Carlo Casino on 18 August 1913. On one roulette table the ball landed on black again and again. As the run grew, gamblers crowded round and staked more and more money on red, reasoning that a streak that long simply had to end.

It did end, but not until black had come up 26 times in a row. By then players had lost millions of francs. The chance of 26 blacks in a row on a European wheel is about 1 in 137 million, which is extraordinary. But on every single spin of that run, red was no more likely than it had been at the start of the evening: 18 pockets out of 37.

Two questions that sound alike

The confusion comes from mixing up two different questions. Before you start, the chance of six heads in a row is 1 in 64, which is unlikely. So after five heads it feels as though a sixth would be completing something improbable.

But the five heads have already happened. Their probability is no longer 1 in 32; it is 1, because they are in the past. The only uncertain thing left is one flip, and one flip is 50/50. The sequence heads-heads-heads-heads-heads-tails is exactly as unlikely as six heads in a row. It just does not look as remarkable.

But does it not even out in the long run?

It does, and that is exactly why the fallacy is so tempting. The law of large numbers says that as you flip more and more coins, the share of heads gets closer and closer to one half. People take that to mean the coin must correct an imbalance.

It corrects nothing. The imbalance is not cancelled; it is diluted. Suppose you start with ten heads in a row. Heads leads by ten. Flip another thousand times and you would expect about 500 of each, so heads still leads by about ten, but the share of heads is now 510 out of 1,010, which is 50.5%. After a million more flips the share is 50.0005%. The lead of ten never went away. It was swamped. You can watch this happen on the coin flip page, where a chart tracks the share of heads as your flips add up.

Streaks are normal, not suspicious

Part of the problem is that people expect randomness to look tidier than it does. Long runs feel like a sign that something is wrong. In fact they are close to guaranteed.

The longest run in a hundred flips is seven on average. So when a wheel lands on the same name twice, or a coin gives five tails in a row, you are looking at ordinary randomness. A sequence with no streaks at all would be the suspicious one. Our guide asking is a coin flip really 50/50 has more on how real coins behave.

When the past really does matter

The fallacy applies only to independent events. Plenty of situations are not independent, and in those the past is useful information.

• Drawing without replacement. Deal four aces from a deck and there are none left. Card counting in blackjack works for this reason: the deck does have a memory. The same is true of a wheel in Elimination mode, where each winner is removed, so everyone left really is more likely to be picked next time.
• A device that may not be fair. Twenty-six blacks in a row is not a reason to bet on red. If anything it is a faint reason to wonder whether the wheel is biased towards black. Evidence of a streak can only ever point towards more of the same, never towards the opposite.
• Skill and form. A basketball player's shots are not coin flips. For decades the "hot hand" was dismissed as an illusion, but more careful analysis since 2018 suggests that a modest hot hand is real. People have skill, confidence and fatigue. Coins do not.

How to stop falling for it

Ask one question: does this thing have any way of remembering what happened before? A coin, a die, a roulette wheel, a lottery machine and a random wheel on this site do not. Each spin here is drawn fresh from a secure random generator, with no record of earlier results feeding into it. If the answer is no, the last result tells you nothing about the next one, however long the streak.

Then be honest about the feeling. The sense that something is due is strong and it does not go away just because you know better. Casinos rely on it, which is why roulette tables display the recent results. Treat that display as decoration. If you would like to see the idea tested on something you control, open the Odds tab on any wheel and run the fairness test: every entry comes out at its fair share, streaks and all. For a different way that intuition misreads chance, try the Monty Hall problem.

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