Why doesn't randomness look random? Flip a fair coin 100 times and the longest run of one side is usually 6 or 7 flips long. People who invent "random" results avoid such runs and switch sides too often. Here you test it on yourself, on 100,000 simulations and on the history of the gambler's fallacy.
Half the class really flips a coin, and the other half invents the results and writes them down so they "look random". The teacher glances at the two sheets and almost always spots the fake. How?
The coin is fair: heads and tails each come up with probability 1/2. Flips are independent, and the coin has no memory of earlier results.
A run is a stretch of identical results in a row: HHHHH or TTTT. We care about the longest run of either side in 100 flips.
A switch happens when neighbouring flips differ: tails after heads or the other way round. 100 flips have 99 neighbouring pairs.
How long is the longest run in 100 fair flips, typically? How many switches are there? And does tails become more likely after five heads in a row?
"Six heads in a row? That's suspicious"
In fact 100 flips contain a run of 6 or more 80.7% of the time, and the longest run averages about 7. Real randomness looks clumpy. What gives a fake away is how even it is.
If you count runs of heads only, the longest is about one shorter on average: around 6.
Lock in your answer before reading on. What longest run of one side do you expect in 100 fair flips?
Your answer will be compared with the exact distribution here.
Press "Heads" and "Tails", or type H and T in the box, until you have 100 "random" results. Meanwhile the page flips a real coin 100 times. The detector sees both sequences without labels and guesses which one a person made up. It looks at just two things: the longest run and the number of switches.
Theory for 100 flips: the longest run averages 6.98, is most often 6, and has a median of 7. Runs of 6 or more are highlighted.
The computer flips a coin millions of times, splits the flips into sequences and finds the longest run and the number of switches in each. The line is the exact distribution, computed by dynamic programming; the dots are the simulation.
To make the longest run one flip longer you need roughly twice as many flips. The flips axis is logarithmic. Each dot is 2,000 simulated sequences of that length.
No single flip knows about the earlier ones. But a run can start in many places, and 100 flips give many such chances. Here are five ways to see it.
The gambler's fallacy is the belief that after a run of heads, tails is "due" because chance supposedly evens out. Six heads in a row really is unlikely, 1/64, but only before you start. Once five heads are in, the sixth flip is heads or tails with the same chance of 1/2.
Any particular six flips are all the same with chance 2/64 = 1/32. But 100 flips contain 95 windows of six in a row, and each of them could become a run. Many small chances add up to a big one.
Amos Tversky and Daniel Kahneman called this "belief in the law of small numbers": people expect even a short stretch of a random sequence to resemble the overall picture. That is why invented sequences are too even in short stretches: heads and tails alternate in them more often than in real ones.
Each pair of neighbouring flips differs with probability 1/2, independently of the other pairs. So the number of switches is binomial with n = 99 and p = 1/2. A real sequence has 60 or more switches only 2.2% of the time. People inventing flips often switch far more than that.
The recursion's logic: the last run of the sequence has some length j from 1 to k−1, and before it sits a shorter valid sequence ending in the other side. The side of that last run is then fixed. For 100 flips and k = 6 this gives exactly 80.68%. The page computes all these numbers this way, with no approximation.
The gambler's fallacy can be checked on a million flips. Then we look at roulette with a zero, and at one of the subtlest stories in statistics: how a study of the "hot hand" in basketball turned out, 33 years later, to contain a hidden bias.
The simulation flips a coin a million times and, after every run of k heads, looks at what came next. Each press adds another million flips.
A wheel has no memory, just like a coin. But on roulette red is not "half": the green zero takes part of the odds. A bet on red after any run of blacks wins with the same chance, 18/37 on a European wheel and 18/38 on an American one.
So a long run of blacks gives no edge at all, and the house edge stays fixed: 1/37 ≈ 2.7% of every bet on a European wheel.
In 1985 Gilovich, Vallone and Tversky tested whether a basketball player hits more often after several hits in a row. For each player they took the share of hits after a run of hits and compared it with the share after a run of misses. They found no difference, and the hot hand was declared a cognitive illusion. In 2018 Miller and Sanjurjo showed that this way of counting is biased: even for a fair coin, the share of heads after heads, averaged over players, is below 50%.
Where does the bias come from? Each player gives a single number, however many runs he had. Sequences with long runs have many shots "after heads", almost all of them heads, yet such a sequence weighs in the average as much as a sequence with a single run. Pool all the shots instead and the share is exactly 50%. Once Miller and Sanjurjo corrected for this, they found signs of a hot hand in the controlled shooting experiment from the original paper.
At the Monte Carlo Casino a roulette wheel came up black 26 times in a row. Gamblers kept betting more on red, sure it was "due", and lost millions of francs. That is why the gambler's fallacy is also called the Monte Carlo fallacy.
The chance that 26 spins of a European wheel, starting from a given one, all come up black is (18/37)²⁶, about 1 in 137 million. But thousands of wheels spinning for decades add up to a huge number of spins, so a run like that turns up somewhere sooner or later.
That was the title of a 1971 paper by Amos Tversky and Daniel Kahneman showing that even scientists expect small samples to resemble the whole population too closely. Kahneman later received the Nobel Prize in economics for research on errors of judgement.
Thomas Gilovich, Robert Vallone and Amos Tversky analysed shots by Philadelphia 76ers players and ran an experiment with Cornell University basketball players. They found no hot hand, and for decades the paper was the classic example of people seeing patterns in randomness.
Joshua Miller and Adam Sanjurjo published a paper in Econometrica showing that the 1985 method systematically understates the share of hits after a run of hits. After the correction, data from the same experiment point towards a hot hand. The debate about its size goes on.
Statistics teachers have long shown this trick. One group flips a coin, the other makes up the results. The teacher finds the fake in seconds: the invented sequence has no long runs, and heads and tails alternate too often.
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