From the moment of the National Lottery's inception in November 1994, it has been the most recognised consumer brand in the UK. Thirty million people - 68 per cent of the adult population - regularly buy tickets, spending an average of £3.32 a week, plus another £1.50 on instants, at the 27,180 lottery terminals installed around the country. And it is so addictive that 99 per cent of jackpot winners continue to play after making their fortunes.
Few of the arguments that made the Lottery's launch so controversial have abated. Supporters still see it as a harmless piece of fun for the whole country that makes a lucky few extremely rich and raises phenomenal sums of money to help good causes. Its detractors see it as a pernicious exploitation of the gullible, a tax on innumeracy and hope. But after five years, surely we can do better than re-rehearse such futile arguments? Instead of debating the relative merits of good v evil, let us consider the Third Way, in which, viewed dispassionately, the Lottery can become an exercise in applied probability theory which could even lead to profits for an astute punter.
On the face of it, this seems implausible. Everybody knows that the Lottery is a mug's bet. You buy a ticket for £1, of which only 50p is returned in prizes. Almost any other form of gambling will supply a better return on investment.
Some people try to improve their chances by studying the form of the 49 numbers, betting on those that haven't been picked for a long time; others do the exact opposite, picking numbers that have come up frequently. Such views are equally deluded: in the behaviour of a random variable, past form is no indication of future performance.
Yet there is a way, in theory at least, that one could make a profit on the Lottery, and an analysis of the results so far suggests that it might just work. The plan, essentially, is to win the jackpot - ideally in a rollover week - and not share it with anyone else. Sounds good, doesn't it? Now here's how you do it.
The principle is to bet only on sets of numbers that nobody else has chosen. That way you will ensure that when you win the jackpot, you won't have to share it. With £50 million-worth of tickets sold for the Saturday draw and £25 million for the midweek one, you might expect almost every combination of numbers to have been chosen several times. There are, after all, only 13,983,816 different ways of picking six numbers from 49, which is nowhere near enough to go round. However, the huge variance in numbers of jackpot winners - from zero to 133 at one go - confirms that some combinations are far more popular than others. So all we have to do is ask Camelot for a list of all the combinations that nobody is betting on, buy up all those tickets and wait for the money to roll in.
That is where our plan hits an important hitch. While the Camelot computer does keep a record of all tickets purchased, the details are a matter of the utmost secrecy. This is not a terminal difficulty, though. With the results of nearly 400 draws behind us, we can reach enough conclusions to increase our winnings. The crucial pattern was already established in the early months of the lottery. Compare these three draws: week three (no winners): 11, 17, 21, 29, 30, 40; week eight (no winners): 2, 5, 21, 22, 25, 32; week nine (133 winners): 7, 17, 23, 32, 38, 42. From these alone, we can form three hypotheses:
First, birthdays have nothing to do with it. When the Lottery was born, there was a common belief that people tended to bet on birthdays, which would make the numbers from 1 to 31 more popular than the rest. Yet the hugely over-subscribed jackpot contained three numbers in the 32-49 range, while the others had only one each.
Second, people don't bet on adjacent numbers: the draws including the pairs 29-30 and 21-22 didn't produce winners.
Third, people do bet on numbers that look well spread: the 7, 17, 23, 32, 38, 42 winning line not only contains no adjacent numbers, but contains representatives from each group of ten: the units, tens, twenties, thirties and forties.
The latter two hypotheses have stood the test of time. Let's first take the draws that have produced pairs of consecutive numbers. One would expect this to occur just under 50 per cent of the time. In fact there have been only 170 such draws, with 226 producing no two adjacent numbers. But those 226 where the numbers were spread out have had an average of five people sharing the jackpot, while the 170 draws that produced pairs had an average of only 2.6 winners. So the share of jackpot prizes received by each winner in an adjacent-pair week must have been almost twice as much as in non-pair weeks.
If a pair of numbers doubled our winnings, what, we might greedily ask, will two pairs or three-in-a-row do for us? The answer is surprising: it makes things worse. The average number of jackpot-sharers when the lottery draw produces two pairs or three-in-a-row is three, which is rather worse for the winners than when only a single pair emerged. Draw number 87, for example, which produced the splendidly huddled pattern of 5, 10, 11, 12, 41, 42, led to no fewer than 12 winning tickets.
The most likely explanation is that a large number of people are deliberately betting on such patterns because they know that most people like their numbers well spread. One of the few pieces of information that Camelot has released about betting patterns is that every week 13,000 people pick the combination 1, 2, 3, 4, 5, 6 - paradoxically because they think it highly unlikely that anyone else will pick it.
The well-spread theory is even more strongly supported by the data. One would expect roughly one in seven draws to produce the well-spread pattern of one ball in each of the 1-9, 10-19, 20-29, 30-39 and 40-49 ranges, plus one more to make up the six. Looking at the draws that have produced more than ten winners, however, shows a surprising proportion fitting this unusual pattern. There have been more than ten winners on 20 occasions, of which 12 conformed to our well-spread criteria.
Camelot appeared to make things more difficult for us when it introduced the lucky dip option. With 28 per cent of ticket-buyers now choosing to have their numbers randomly allocated, one would expect less bias to be evident in the choices, yet the statistical variations mentioned above seem just as strong as they were before the lucky dip. It looks as though the 72 per cent who are determined to pick their own numbers are even more biased than the rest, so we still stand a chance.
Using the above findings, we can generate sets of numbers that stand a good chance of not being duplicated by too many other punters. We'll avoid all sets which have something in every batch of ten; we'll avoid anything that doesn't have a pair and anything that has three-in-a-row or more than one pair. Our sets of numbers will thus be weird, but not ostentatiously so.
Pick a million or so such combinations and bet on all of them every time a rollover arrives. You can then expect one major jackpot, unshared, every 14 times you enter, which, with rollovers about one every 12 draws, will be more than once every two years. All you need is the initial stake of £14 million and you should turn over a healthy profit. With a bit of luck.
The writer is author of "The Book of Numbers", a new edition of which is out in March 2000, from Metro