Blog: Plutonium

Bruce Springsteen sang, on his Nebraska album, "now mister, when my number comes in, I ain't never gonna ride no used car again".

Leaving aside the semantic issues of the triple negative, this illustrates a contrast between the UK and US lottery systems. In America, each ticket has a single serial number, and whichever serial number is selected, is the winning ticket. Hence "when my number comes in", rather than, for the UK, "when my numbers come in". In the Uk, we have 6 numbers between 01 and 49, and it is the combination of these numbers that wins. As a corollary, there are (as I understand it) no minor prizes in the US lottery; it is 'winner takes all'.

In the UK lottery, there are 49 balls. The chances that the first ball that comes out, matches one on your ticket, isn't too bad. 6/49 or 0.1224 - roughly 1 in 8. However, as progressive balls come out, and assuming each ball matches a number on your ticket, the chances for the next ball decline rapidly. On ball 2, there are 48 balls left, and 5 remaining numbers on your ticket. That's 5/48 or 0.101. Then 4/47 for ball 3 (0.085), 4/46 for ball 4 (0.0870), 3/45 for ball 5 (0.066), and 2/45 for ball 6 (0.044).

All these are low but nevertheless realistic probabilities. However, your overall chances are obtained by multiplying all these figures together. This comes to 720 / 10,068,347,520, or 1 in 13,983,816.

When I ran a lottery syndicate, members kept asking me "when are we going to win the biggee?"

The Maths for this turn out to be quite simple. If for example, I play a dice game in which I win if I throw a 6, I can expect to wait on average 6 throws, to win for the first time. This is not hard and fast, of course. I could throw a 6 on the first throw, or I could still be waiting for my first 6 on my 20th throw. This is the geometrical distribution, with a peak at the beginning but also with a long 'tail'.

So a person buying one lottery ticket a week can expect, on average, to wait 13,983,816 weeks, or 268,919 years, for their first 6-number win.

Again, this is not hard and fast. You could win next week, or you might still be waiting 2,000,000 years in the future.

Obviously, you can improve your chances by buying more tickets. But how many would you have to buy, to ensure an average wait that is within a realistic human lifetime?

For an average wait of 50 years, or 2,600 weeks, you would have to buy 13,983,816 / 2,600 = £5,378 worth of tickets, each week.

Of course, the corollary of this is that, over that time, you actually spend £13,983,816, so you would need to win at least this much, to make it worthwhile.

Good luck if you do play.