An uncommon level of interest has been generated by the National Lottery, an enjoyable mechanism whereby money is painlessly extracted from The Masses, and re-distributed to The Few. One of the early winners received 18 million pounds - the equivalent of 30p for every man, woman and child in the United Kingdom.

The possibility of achieving such an outcome for a modest outlay inevitably invites comparison with other methods of spending one's grant. The tickets cost £1 a throw, which is less than half the price of a pint of bitter, and about equivalent to buying two issues of The Times. For that amount you can purchase the opportunity to have a shot at winning a prize of enough magnitude to take over a newspaper or (even better) a brewery of your very own.

A statistician analysing the lottery would be interested in the probability of winning, which has been calculated at one in about 14 million. (For comparison, that is about twice the population of Greater London.) Since the sum of all probabilities must add up to one (or 'unity', as they say), this means that the probability of not winning is 13,999,999 in 14 million.

An investor wondering whether to take part in the lottery would be interested in the 'Net Present Value' of investing in a single lottery ticket, or NPV. Following on from the above, and ignoring for now the possibility of multiple winners, the NPV would appear to be the size of the kitty multiplied by one in 14 million (see above), less the outlay of £1.

The question is, Can the punter move either of these factors in his favour? We will begin with the investor's viewpoint, as it is easier to analyse.

1) Net Present Value

The first question is, can the investor's likely return, the NPV, be increased? And the answer would appear to be yes. In order to increase one's NPV, various rules have been suggested. Of the five cited below, the first is clear, while the next three seem to be sound enough, and can be used together. The fifth is dubious.

Rule number 1: Play only on roll-over weeks, when there is more money in the kitty. The NPV is, as mentioned above, the size of the kitty, multiplied by one in 14 million, less £1. If the kitty rises to over £14 million, then the NPV on each ticket becomes positive. (As an aside, it would then even be a rational strategy to buy all 14 million tickets. However, it would not be safe to do so, as it ignores the very real possibility of having to share the winnings.)

Rule number 2: Prefer the high numbers. Low numbers are chosen by very many punters, especially those who choose numbers with a meaning, such as someone's birthday. Street numbers also produce a bias towards low numbers. Ergo, avoid them if you do not want your winnings to be correspondingly divided by the number of winners. Not that I would mind sharing first prize with someone...

Rule number 3: Avoid favourite numbers. In particular, never choose the numbers 3, 7, 11 or 13.

Rule number 4: Do not go for a pattern. For example, avoid choosing one of the multiplication tables, square numbers or bits of the Fibonacci sequence. This is not because they are any more unlikely to come up in the draw than any other sequence, but rather because many people go for patterns, and this tends to increase the chances that you will have to share your winnings.

Rule number 5: Choose three even numbers and three odd. The reasoning behind this is the 'law of averages', which suggests that the
winning selection of numbers will usually consist of this combination. This is, in fact, an example of a rule that should not work. It is possible that it might even decrease the NPV, as choosing three of each is in fact a pattern, albeit a low-level one.

2) Probability of winning

The probability of winning on each ticket is fixed. But what about the lifetime probability? The question is, can you improve the lifetime probability of winning by buying all one's future tickets at once rather than in the future - or vice versa?

If you assume a fifty year playing time, with 50 weeks to a year, the result is 2,500 weeks. Thus you can base the maths around buying 2,500 tickets, each with a probability of winning of around 1 in 14 million. If you buy them all now, assuming they are all different, the odds of winning are 2,500 in 14 million, which is 1 in 5,600. This is still, incidentally, vanishingly small.

If you buy the same tickets one per week, spread over 2,500 weeks, the odds of NOT winning each week are 13,999,999 divided by 14 million, and the cumulative probability if NOT winning, over the course of the period, comes to 13,999,999 divided by 14 million, all raised to the power of 2,500.

When you do the number-crunching, you find that the odds of winning over a lifetime do in fact come to almost the same figure as before: 1 in 5,600 - in fact the chance is very slightly less, but the difference is insignificant. (This calculation approximates so closely to the previous one because 13,999,999 in 14 million is so close to one, even with 2,500 iterations.)

One could summarise the comparison by saying that, if you buy the tickets one per week instead of all at the same time, there is a slightly higher chance of winning nothing, but that is counterbalanced by an infinitesimally tiny chance of winning twice (or more!) over the period - something that is obviously impossible if you buy all the tickets at the same time.

Thus it seems that the lifetime probability of winning the lottery cannot be increased by buying all your tickets at once. Of course, such a method might set your mind at rest once and for all, win or lose...

(Note: This piece was inspired by another entry, by Sir Kitt.)