In August 2010, a newspaper that’s better at being outraged than being accurate claimed that thousands of people who played the Lotto Thunderball game were being unfairly prevented from winning. Until May that year, players had chosen five numbers from the range 1 to 34; in May this range was extended to 1 to 39, but some shops were discovered to be using old forms which didn’t allow players to choose numbers 35 to 39. Players who’d been sold the old forms therefore had no chance of winning on any occasion when one of these higher numbers came up, and the newspaper trumpeted that “Since May, the five numbers have come up 24 times in the three-times a week draw — eight times in the last month alone”. Consumer representatives were “shocked” and “astonished”; an altruistic lawyer suggested that the Lotto operator Camelot could be sued; Camelot opened an inquiry and traded recriminations with the shops; and politicians rushed to join in the condemnation.
Buried in the middle of the story, and apparently ignored by everybody else, was a comment from a mathematician who explained calmly that being sold an old form “makes no difference to your chance of winning”. So was he right and was everybody else wrong?
For those of you who have enough sense not to be familiar with the rules of Thunderball, here’s how it works. You pick 5 natural numbers from the range given, i.e. [1,34] in the old version and [1,39] in the new version. Additionally, you pick a natural number from [1,14] as the “Thunderball”. There are various minor prizes, but the main one is a fixed prize of £500 000 which you will win if all six of your numbers match those chosen in the draw. (If several people match the numbers correctly they each receive the full prize.) We presume that the draw is random and that each number has an equal chance of being drawn.
Assuming that you can choose from [1,39], your probability of winning is , where is the number of possible ways to choose five distinct numbers from [1,39] and one number from [1,14]:
We can also calculate how likely it is that at least one of the numbers [35,39] will come up. This is simply , where is the number of ways to choose your numbers from the range [1,34]:
So more than half the time the old form has no chance of winning — a scandal, surely?
Well, no. As our own Dr Phil Knight correctly pointed out to the newspaper, any selection of five numbers chosen from the old range has exactly as much chance of winning as any selection of five numbers chosen from the new range — it’s still just a set of five numbers within [1,39], which is as likely to come up as any other set of five numbers is. This may seem counterintuitive, so consider a more extreme case. Suppose that the shop instead sells you a form on which the numbers are predetermined, like a raffle ticket. You now have no choice of your numbers at all, but the set you are given is still as likely to win as any other set of numbers you could have chosen! (That is to say: if any other numbers come up in the draw then you had “no chance of winning” so you ought to sue Camelot… right?)
It’s interesting that this fact really offends our intuition: it’s very hard to believe that in a completely random game having our choice restricted makes no difference. This seems to be consistent, though, with a well-known principle in risk perception: our intuitive assessment of risks is strongly affected by how much control we feel we have over a situation. Notoriously, many of us get more worried about the risks of flying (which is very safe, but where as passengers we clearly have no control) than about the risks of car travel (which is a lot more dangerous either per hour or per km). If we get to make a choice — even one that has no effect on the outcome whatsoever — we feel that the outcome does depend on our decision; thus we are in control and we overestimate our chances of winning or underestimate the chances of losing. This, of course, is precisely the psychological flaw that lottery operators exploit to encourage us to play!
By the way, based on the odds on the Thunderball page, your expected return for a £1 ticket is just under 53p. If the thrill of believing briefly that you might win is worth more to you than 47p, you can probably make a decent argument for playing the game. If you get a kick out of gambling, though, learning to play poker well would probably still be a more interesting and profitable use of your time…