Randomness is clumpy

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The above screenshot shows an interesting point in a sequence of games of roulette. The results of the previous nine spins are shown at the bottom of the screen in reverse order, so black 17 was the most recent.

Above this we see the bets that have been made on the next spin. Note the huge stack of chips on red (circled) compared with the tiny stack betting on black. The gamblers clearly believe that a red is “overdue”, since the last seven spins have all been black. This is an example of the so-called gambler’s fallacy. Probability theory tells us that a black is equally likely on the next spin.

So what did happen next?

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Not only was the next spin a black, it was black 17 again. In fact, black 17 had come up three times in the last nine spins.

This is a great example of how clumpy randomness is. People tend to associate randomness with evenness. In the very long run they’re right. In a very large number of spins, you would expect to see black about half the time, and black 17 about one time in 37. But in the short run, you often get clumpy results such as this.

It’s because clumpiness doesn’t feel random that Apple had to fiddle the shuffle feature in iTunes to make it seem random by avoiding playing two songs by the same artist one after the other.

An exercise I like to use with students is to ask them to write down a sequence of 100 random digits generated from their own heads. Two things typically happen. First, they find it surprisingly hard. Initially they write their digits quite quickly, but they soon slow down. This is because they’re thinking: they’re trying to make the digits look random. (Sometimes they just give up and start writing down sequences they already know, like phone numbers.)

But the second thing that happens is that they fail. For example, about one time in ten you would expect the same digit to be repeated. So in a list of 100 random digits, you’d expect about ten repeats. Typically students will generate fewer than this. You’d also expect to see one example (on average) of three of the same digits in a row in a set of 100 random digits. In a class of students, this very rarely happens. Indeed, in a class of 30 students, you’d expect to see about three examples of four identical digits in a row. This never seems to happen because it just doesn’t feel random.

But it is. Randomness is clumpy.