Mathematician Ian Stewart explains why the National Lottery will almost certainly be won by somebody else.

As a mathematician I have an ambivalent attitude towards that instant British institution, the National Lottery. It is a standing tribute to public innumeracy and a fascinating example of probability theory in its simplest and purest form - gambling. I do not gamble myself, but I have no moral objections to it, although I do not see why the Government permits unscrupulous operators to advertise and sell "systems" that allegedly enhance your chances of winning but are wholly without merit. I also think it is time we had some "truth in advertising" laws, so that "It could be YOU" is always accompanied by "Government Wealth Warning: it will almost certainly be SOMEBODY ELSE", and "there's a winner every second" is juxtaposed against "there's one born every minute".

Games of chance are governed by rigid mathematical laws, the laws of probability. These have been honed over centuries and subjected to stringent tests. Moreover, the concept of probability originally derived from gambling. In 1654 the Chevalier de Mere wrote to his friend, Blaise Pascal, asking how the stakes in a game of dice should be divided if the fuzz raided the joint and interrupted it. Pascal, who, ironically, became a great moralist, invented much of the basic machinery of probability theory to answer him.

Probability is a numerical quantity associated with random events. The so-called law of large numbers lets us interpret probability as frequency, so that in the long run the probability of an event is the proportion of occasions on which that event occurs. The lottery involves the random selection of six balls, plus a bonus ball, from a total of 49. The probability of selecting any given ball is 1:49, and each ball is exactly as probable as any other.

Many misconceptions about probability hinge on the law of averages, which asserts that any unevenness in random events gets ironed out in the long run. There is a sense in which this is true, but the way people usually interpret it is wrong. Think of tossing a coin. Tails do not become more likely if you toss surplus heads; neither the coin nor its surroundings "remember" what happened on previous occasions. Suppose you happen to toss ten heads in the first ten trials, a frequency of 1, very different from the typical value of 0.5. Succeeding tosses are not biased in favour of tails: instead, extra trials yield roughly equal numbers of each. Suppose, for instance, that in a further thousand trials we get 500 heads and 500 tails. Then the total number of heads is now 510 out of 1010, and the frequency is 510/1010 = 0.505 which is a lot closer to 0.5. So later trials just swamp any imbalance by weight of numbers.

This means that all lottery systems based on analysing past draws - such as elaborate computer programs - are totally useless. The more clever the pattern-detecting software is, the more cleverly it is analysing the wrong thing. The patterns it thinks it finds are spurious coincidences.

Without the jackpot nobody would play, so for simplicity I will ignore the other ways to win smaller sums. What is your chance of winning the jackpot? Let us eavesdrop on the British public as the numbers are drawn. Here comes the first ball: all punters who did not choose that number are immediately out of the running. There are six ways to be right out of 49 choices, so on average only 6/49 of the population remains in the game. One reason for betting on the lottery is the excitement of the draw - well, roughly six people out of seven get very little excitement. Here comes the second ball: surviving punters have five chances out of 48 of getting this one right (they have five choices left and one ball has been drawn already). Now only one person in 80 has any interest in the jackpot. Ball three reduces the interested population by a factor of 4/47, and one person in 921 survives. So halfway into the draw, of the 20 million watching, only 20,000 still have jackpot hopes. Ball four reduces this by 3/46 and we are down to one person in 14,125; ball five cuts the numbers by 2/45 and only one player in 317,814 remains. Finally the sixth ball reduces the survivors by 1/44 and only one person in 13,983,816 is left.

That's your chance of winning: roughly one in 14 million. Yeah, sure, but it's a big prize, innit?

Statisticians measure the fairness of a game of chance by calculating your expectation - the amount you win on average. In a fair game your expectation is zero. What is it here? The typical jackpot is around Pounds 8 million and you bet Pounds 1. On the simplifying assumption that at most one person wins the jackpot, which overestimates your likely winnings, your expectation is a loss of 43p. In gambling terms, the lottery is a "sucker bet". Of course there are the other prizes, but your expectation is still a substantial loss.

One implication of negative expectation is that the more you play, the more you probably lose. Many people will try to sell you a system that needs lots of tickets - say a way to choose eight numbers with a guarantee that if at least five of those are selected then you will win the jackpot - which as it happens requires exactly 2,464 tickets. Purveyors of such systems hope you do not notice that they only work if you get those five numbers, and that is very unlikely. Indeed, its probability neatly balances out, so that your chances of fulfilling this condition are the same as your chances of winning the jackpot if you place any 2,464 different bets.

Although the lottery is a sucker bet, that does not mean it is not worth your while to play. The typical punter bets only a small sum every week, and for them the expectation gives an incomplete picture. Most pour small sums down the drain - about Pounds 100 per year - hoping for numbers that never turn up. Occasionally they win Pounds 10, but that is just a psychological sop to keep them interested, and it is more than cancelled out by their losses. What they buy is Pounds 100-worth of excitement and a possibility - albeit remote - of instant riches.

The pattern for the very few who hit the jackpot is utterly different: what they buy is a change of lifestyle. So should a "rational" person play? There are many considerations, among them: "do I really want a total change of lifestyle?" The mathematically minded must beware of their own fallacy, for it can be rational to bet when your expectation is negative. The simplest example is life insurance, in which players hope not to hit the jackpot. Players pay an affordable premium to purchase financial security in the event of an unlikely disaster, and this is perfectly rational behaviour.

Now we have finally reached the bit you have been waiting for. A mathematician is going to tell you how to win a fortune on the lottery. Come on, get real. If I knew how to win, do you think I would give the secret away for the price of a newspaper?

No system can improve your chances, which are determined once and for all by the laws of probabilty. But an intelligent system can improve your expectation. If you must play, the smart move is to make sure that if you win, then you win a bundle.

One tactic is to bet only when there is a rollover. Another was made dramatically clear when 133 people won a jackpot of Pounds 16 million and got only a miserable Pounds 122,510 each. Never bet on numbers anybody else is betting on. How can you do this when you do not have access to the Camelot computer? Avoid having too many numbers under 31 because of all those birthdays that everybody falls back on when asked to choose numbers. Avoid three, seven, 17, and other numbers people think lucky. You could sprinkle in a few 13s: those little plastic balls are not superstitious. I recommend something really stupid like 43, 44, 45, 46, 47, 48.

But so many people now wait for rollovers that you no longer gain a useful advantage by doing so unless you can guarantee choosing numbers nobody else will come up with. Since many people choose their numbers using tiny random number generators, this is getting extremely difficult, and it leaves only one surefire tactic - get lucky.

Ian Stewart is professor of mathematics at Warwick University. His new book, From Here to Infinity: a Guide to Today's Mathematics,was published yesterday by Oxford University Press.

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