Do you rely on a dead priest to choose your lottery numbers? Ian Walker assesses the statistics thrown by the nation's biggest draw
The National Lottery has attracted billions of pounds for good causes. What it has not attracted is any serious analysis of why people like it, whether the way in which the game is structured is the best way, and whether it was desirable to set up the lottery in the first place.
The topic is important - more important than simply the amount of money at stake (large though that is). There are many circumstances where individuals make decisions when there is risk - yet we have little good data, so the opportunity to analyse lottery-ticket purchases is a valuable experiment.
Much economic analysis presumes that behaviour is determined by the interaction between individual tastes and the prices and incomes they face: if lottery tickets were a better buy more would be sold, and it is the design of the lottery that determines how good (or bad) a buy lottery tickets are. An investment that yields a return of about minus 50 per cent a week needs to be a lot of fun to tempt people to invest, but the behaviour of people who play for fun is still sensitive to how much that fun costs.
For those still not versed in the game, players pay Pounds 1 for the privilege of choosing six numbers between one and 49. The chances of matching the six drawn (more or less at random) is about one in 14 million, so, with sales in a typical Saturday draw of around 60 million, we initially expected, when we started our research, to get about four jackpot winners a week. The chances of matching three balls is about one in 57, so we should see a little over one million Pounds 10 prizewinners each Saturday. Indeed with 60 million players the law of large numbers implies that we should get fairly close to a million small winners and four jackpot winners sharing about Pounds 12 million (52 per cent of 45 per cent of what is left from the Pounds 60 stakes after the million or so Pounds 10 winners, where 52 per cent is the jackpot's share, 45 per cent is what is returned as prizes in total).
The first surprise in the research was that the game in practice is nowhere near so predictable as we had initially thought. While the average number of prizewinners of each type conforms to the simple statistical picture, the variance around this is much larger than you would think. We expected, with weekly draws and 60-70 million players, to get very few rollovers (when there are no jackpot winners) and no occasions when the number of jackpot winners reached double figures. In fact there have been nearly ten times as many rollovers than we had expected and the number of jackpot winners has twice exceeded 50.
These surprises arise from the fact that the number of tickets sold does not reflect the proportion of the 14 million combinations of numbers chosen. Players do not choose their numbers randomly: lucky numbers, birthdays, "hot" numbers and combinations of numbers that make patterns on the ticket are important features of behaviour. No matter how bizarre your method of choosing numbers, you can be reasonably sure that, with 60 million players, there will be others who "share" your method. A large Irish jackpot was recently shared by two winners who both said that their choice of numbers came from the birth, death and ordination of a particular priest.
This behaviour generates more rollovers and in a rollover week a lottery ticket is a better buy than in a regular week. But it also means that tickets are worth a bit less in non-rollover weeks than they otherwise would be, since there is a bigger chance that the jackpot will disappear from the expected return in such weeks. The bigger this chance the lower the return on buying a ticket. Our research focused on working out how the demand for tickets varied according to the potential return.
To work this out we needed to know the distribution of numbers people pick. Unfortunately this is commercially sensitive information. But some devious statistical analysis of published data enabled us to make some estimates. We estimate that the least popular numbers are 36, 41, 46, 47, 48, 49, while the six most popular are 7,17,18,19,23,. Betting on the former would yield a return (in the very long run) of about 11 per cent a week (not many investments yield this); betting on the latter a loss of about minus 78 per cent.
The reason for estimating this is not to make money but to measure how sales change when the return rises. Our measure implies that a 1 per cent increase in the return induces a 1 per cent increase in sales - that the price "elasticity" of sales is unity. This is precisely what would be implied if the game had been structured to maximise sales.
Our estimate also implies that the game is a lot of fun; if it was not, then demand would be unresponsive to price changes; compare boring things like domestic electricity consumption. It also implies that taxing this fun reduces it a lot. So imposing a 28 per cent levy for good causes spoils a lot of fun. It may be better to stop raking 28 per cent off from sales, offer more prizes and simply confiscate the jackpot whenever a rollover occurs.
Least popular numbers: 36, 41, 46, 47, 48, 49
Most popular numbers: 7, 17, 18, 19, 23,
Ian Walker is professor of economics, Keele University.