Why does anyone win the lottery?

lottery ballsConsider a lottery in which there are $N$ possible number combinations - in the UK Lottery $N $= 13,983,816. Each ticket therefore has a $1/N$ chance of sharing in the jackpot. Suppose we sell $N$ tickets, what is the chance of nobody winning the jackpot? What if we sell $2N$? The chances are shown in the table below, and hold whatever $N$ is, provided it is large!

Table showing that, if $N$ is large, the chance of nobody winning ranges from 37% if $N$ tickets are sold to 1% if $5N$ tickets are sold.

Chance that nobody wins the lottery if we sell $N$ tickets
Numbers of tickets sold % Chance nobody wins
$N$ 37%
$2N$ 13%
$3N$ 5%
$4N$ 2%
$5N$ 1%

On average around 37,000,000 tickets have been sold for each draw, around 2.6$N$, and so there should be a 7% chance that noone wins and there is a rollover, assuming lottery ticket numbers are chosen at random. Of course this is not the case (unless they use the Lucky Dip option) and people tend to choose particular patterns of numbers that feature birthdates and so on, which would increase the rollover rate. Up to June11th 2008 there had been 1301 draws and 234 rollovers (18%), many more than would be expected if people picked numbers at random.

To see where these numbers come from, see In Maths of coincidence.