What are the chances?

25 September 2004

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Robert Matthews

MANY people would call them spooky: bizarre coincidences that loom out at us from the randomness of everyday events. But obviously there's nothing in them. Everyone knows that randomness is the very essence of patternless, lawless disorder.

Obvious, but wrong. Peer hard enough into the fog of randomness, and you can glimpse regularities and universal truths normally associated with deep cosmic order. Why? Because what we call randomness is only a chained and muzzled version of the real thing. When forced to act within certain limits, imposed on it by the constraints of the world we live in, randomness sheds just a little of its notorious mathematical lawlessness. The effect is often subtle, but sometimes it's as plain as day - when you know where to look.

Take lottery numbers, for example. Surprisingly, barely half of all lottery draws look like the kind of jumble of six numbers one expects when randomness is at work. Among the other half, tiny specks of order appear: a pair of consecutive numbers, perhaps, or longer or more intricate runs.

But no one is fixing the lottery - statistical tests have proved this - so what is going on? Look at the numbers used in each draw. Truly random numbers know no bounds but those in the lottery have no such freedom. In the UK's Lotto game, for example, they are confined to the range 1 to 49. And whenever randomness has its style cramped in this way, with only certain outcomes allowed, it loses some of its utter lawlessness and unpredictability. Instead, it must fall into line with probability theory, which describes the behaviour of infinite randomness in a finite world.

In the case of the UK's Lotto, probability theory proves that examples of apparently anomalous order will show up in roughly half of all draws. When randomness is compelled to scatter surprises among just a limited number of outcomes, we should expect the unexpected.

Take the weekend of 14 and 15 August this year, the first of this year's English football premiership competition, when 20 teams played each other in 10 matches. It turns out that half of those matches featured players sharing the same birthday. A bizarre coincidence? In fact, probability theory shows that when randomness is forced to scatter the birthdays of the 22 players in each match among the 365 days of a year, there is a roughly 50:50 chance that at least two players in a match will share the same birthday. In other words, around half of the 10 matches played on that first weekend should have seen at least two players sharing the same birthday. And that's exactly what happened.

Probability theory also predicted a roughly 50:50 chance that at least one player out of the 230 playing that weekend would be celebrating his birthday on the day of the match. In fact, two were: Jay-Jay Okocha of Bolton Wanderers, and Johnnie Jackson of Tottenham Hotspur.

Looking harder still at randomness reveals more subtle signs of its revolt against constraint. Around a century ago, the statistician Ladislaus Bortkiewicz produced a classic study of fatalities in the Prussian army that highlights a bizarre link between randomness and a universal mathematical constant known as e. This never-ending decimal number, which begins 2.718281..., often pops up in natural processes where the rate of some process depends on the present state of the system, such as the rate of growth of populations, or radioactive decay.

Bortkiewicz's data shows this universal number can also be found lurking in random events, such as the risk of death from a horse-kick. According to the reports, the Prussian soldiers all faced a small but finite risk of death from horse-kicks, amounting to an average of one fatality every 1.64 years. Bortkiewicz found that of the 200 reports, 109 recorded no deaths at all. Now divide 200 by 109, and raise the result to the power 1.64, the average interval between deaths through horse-kicks. The result is 2.71 - within 1 per cent of e.

A fluke? Not at all: it's to do with the mathematics of what are called Poisson distributions. Probability theory shows that e can be expected to pop up when lots of randomly triggered events are spread over a restricted interval of time. The same is true of events spread over a limited region of space: you can extract a value of e from the impact sites of the V-1 "flying bombs" targeted on south London during the second world war. While there were hundreds of impacts, the chances of randomness landing a V-1 on a specific part of the capital were low. And analysing the data in the same way as for horse-kick deaths leads to a value for e of 2.69 - again, within around 1 per cent of the exact value.

It's a similar story with everything from radioactive decay to the rate at which wars break out between nations over the years. In each case, the chances of the event may be low, but there are lots of opportunities for it to happen, and randomness responds by allowing e to inveigle its way into the data.

Cajoling the constants

Suitably cajoled, randomness will also produce values for probably the most famous universal constant of all. Drop a needle carelessly onto wooden floorboards: the number of times it falls across a gap between the floorboards depends on the dimensions of the needle, the floorboards and... p. It appears because of the random angle at which the needle ends up on the floor. Observe a few tens of thousands of such events and an accurate value of p emerges from the randomness.

If you want to try it without the needles, gather together a million pairs of random integers and check whether each pair has any common factor. Multiply the proportion of the total that don't by 6 and take the square root: the result is an impressively accurate value of p.

This same approach lets you extract values for p from the scattering of stars across the night sky. Compare the distance between any two stars on the celestial sphere with that of any other pair. Do this for the 100 brightest stars in the sky, and the common-factor method gives you a "celestial" value for p of 3.12772 - within 0.5 per cent of the true value.

We humans seem to have a penchant for seeing patterns in randomness, from religious figures in clouds to faces on Mars. We're right to dismiss most of them as nothing more than illusions, but sometimes they are real. Anyone who knows numbers can see that the mystics were onto something - there really are patterns hidden among the stars.