And now for something completely random, by ALEX BELLOS



By ALEX BELLOS, Author of Alex's Adventures In Numberland



Humans find the concept of randomness very hard to understand, and this can get us into big trouble. Randomness fools us all



'As humans, when we come across random clusters we naturally superimpose a pattern,' said Alex Bellos

Here's a party trick. Get two friends. One will flip a coin 30 times and write down the sequence of heads and tails. The other will imagine flipping a coin 30 times and also write the sequence of results. They do this in secret, so you don't know who has flipped the real coin and who has flipped the imaginary one. The 'wow' moment comes when you're presented with the two lists of heads and tails and it's usually instantly obvious which is which.

'So what?' you may say.



Well, this simple trick touches on some deep and bizarre mathematical ideas with great relevance to the real world. The underlying message is that humans find the concept of randomness very hard to understand, and this can get us into big trouble. Randomness fools us all - sometimes in entertaining ways, but sometimes with devastating consequences.

Back to the party trick. We have two lists of heads and tails: one from the real coin and one from the imaginary coin. The first list is properly random. The second is a human attempting to be random. You can tell the difference because humans are very, very bad at faking randomness. We just can't do it.

There are different ways to tell the fake and real randomness apart, but the most obvious is to look for runs of straight heads, or straight tails. If one of the lists has a run of five heads or tails in a row, you can be pretty sure that's the real coin. In a list of 30 coin flips you're reasonably likely to get a run of five.

When someone is imagining coin flips, however, they almost never imagine a run of five straight heads or tails. This is because after two or three heads our brains tend to think, 'OK, time for a tails now.' Our brains are implementing an order, a 'coin memory', whereas in fact true randomness has no memory of what came before.

Randomness creates counter-intuitively large clusters, such as a run of five heads in 30 flips of a coin, which leads to unexpected results. For example, how many people in a room does it take for it to be more likely than not that two of them share a birthday? Remarkably, the answer is 23 - far fewer than you might think. This is because randomness clusters birthdays together, rather than spreading them uniformly around the year.

As humans, when we come across random clusters we naturally superimpose a pattern. We instinctively project an order on the chaos. It's part of our psychological make-up. For example, when the iPod first came out and people started to use the shuffle feature, which plays songs in a random order, many people complained that it didn't work. They said that too often songs from the same album, or the same artist, came up one after another. Yet that's what randomness does - it creates counter-intuitively dense clusters.

'We're making it (the shuffle) less random to make it feel more random': Apple CEO Steve Jobs changed the feature on the iPod after complaints from users

In response to complaints from users, Apple CEO Steve Jobs changed the programming behind the feature: 'We're making it (the shuffle) less random to make it feel more random.'



In other words, each new song now has to be significantly different from what came before, so as to conform to our expectation of randomness. Which isn't really random at all.

The illusion that random clusters are an indication of a pattern is seen most strongly in gambling, which is why it's often called the 'gambler's fallacy'. Slot machines, for example, exploit the fallacy ferociously. If, say, there's a one-in-ten chance of winning a payout, then after ten losing bets it's natural to feel the machine is 'due' - that it's been holding back so long it just has to pay out soon. That's one reason you carry on gambling. Yet the reasoning is wrong. On any one bet, the machine is never more likely to pay out than any other.

Conversely, if you go through a winning streak it's difficult to stop gambling, because you intuitively feel there's a pattern behind that streak. Players call slot machines 'hot' if they're paying out more than expected. Yet machines are neither hot nor cold. Winning streaks aren't due to luck or skill. They're random clusters.

The study of randomness teaches us that coincidences are a lot more likely than you might think. Take the Lottery. Germany has a lottery like ours: for each ticket bought there's a one-in-14-million chance of winning.



In 1986 and 1995, however, the same combination of six numbers won: 15, 25, 27, 30, 42 and 48. What can be the chances of that? It seems absurdly unlikely. Well, not really. Between 1986 and 1995 there were 3,016 lottery draws. The chance of two identical combinations in 3,016 draws is about 28 per cent.

But why are we so bad at understanding randomness? It's because when things are random, we have no control over them, and our default reaction is to try to regain a sense of control. We do this by projecting a pattern on them. Having control over our surroundings is a deep urge, as a famous - and rather ruthless - experiment showed in the Seventies.

The research involved elderly patients in a nursing home. Some were given no control over their environments: the arrangement of furniture in their rooms was decided for them, and a plant was chosen and tended for them too. The others were allowed to choose how their rooms were arranged and could choose and tend for a plant themselves.



The result after 18 months was striking. The patients given control over the rooms and plant choice had a 15 per cent death rate. But the death rate for patients who had no control over their rooms and plants was 30 per cent. With control denied to them, they were twice as likely to die.