## What are the chances of someone winning the (UK) National Lottery twice?

#### Pretty good..but it won't be you

As of 6th Jan 2014, there had been 1882 lottery draws and 4503 jackpot winners (giving an average of 2.4 winners per draw).
Assuming that:
• Winners consider themselves lucky and so continue to play (twice a week).
• Jackpot winners live for 30 years (from the date of their first win).
• The overall number of players doesn't decrease (and so the average number of winners each week remains at least 2.4).
• The lottery runs for another 20 years.
• Players choose their numbers independently at random (say by using the Lucky Dip option).
The chances that someone wins the lottery twice before 2034 is greater than 60%.

#### Why it isn't one in two hundred trillion.

In each draw the chance of a single ticket winning is around 1 in 14 million. (Actually it's 1 in 13,983,816 = 49×48×47×46×45×44/6!.)

So if you buy a single ticket in each of two draws then your chances of winning the jackpot on both occasions is 1/14,000,000 × 1/14,000,00 which is roughly one in two hundred trillion.

But that isn't what we are calculating...

#### The calculation

As of 6th Jan 2014 there had been 4503 single jackpot winners who we assume continue to play in each of the 104 draws per year (until they die).

We will calculate the probability of no-one winning the lottery for a second time in the next 20 years.

If there are no second jackpot winners then the number of (single) jackpot winners grows on average by 2.4 per draw and so totals around 7000 by 2024. At this point our assumption about life expectancy kicks in and so the number of (single) jackpot winners now remains roughly constant for the next ten years (actually the number would still grow a little due to the fact that originally there was only a single draw each week, but this only increases the chance of a second time jackpot win occuring so we ignore it).

Let p = 1/14,000,000 be the probability of winning in a single draw (with one ticket). The probability that none of the 4503 single jackpot winners win for a second time on Wed 8th Jan 2014 is:

(1 − p)4503.

Since the number of (single) jackpot winners increases by 2.4 on average per draw (until 2024) the probability that no-one wins a second jackpot before 2024 (during which time 1040 draws take place) is approximately:

p1 = (1 − p)4503×1040 + 2.4×(1+2+...+1039).

Since the number of single jackpot winners remains around 7000 from 2024 onwards, the probability that no-one wins for a second time between 2024 and 2034 is approximately:

p2 = (1 − p)7000×1040.

So overall the probability of no-one winning the jackpot for a second time before 2034 is roughly:

p1 × p2 = (1 − p)13,259,792.

Using the fact that (1 − 1/n)n tends to 1/e as n tends to infinity this probability is less than 0.4 and so the probability of a second time winner is at least 0.6.

#### How reasonable are our assumptions?

Whether winners really continue to play is difficult know. However it is quite possible that even if not all of the winners continue playing, those who do continue buy more than one ticket in each draw and so the total number of different tickets bought in each draw by previous jackpot winners is as large as we estimated. If so, the answer would remain the same.

Although plenty of winners won't live for 30 years many will live for longer, so as long as the average life expectancy of a winner is around 30 years this won't change the answer dramatically.

The assumption that the number of players isn't decreasing is incorrect. I don't know exactly how the number of players has changed but over 2010-2014 the average number of jackpot winners per draw was 1.55 rather than 2.4 (which was the average over the period 1994-2014). However, if you redo the calculation with the assumption that each draw results in 1.55 jackpot winners (on average) you still find that the chance of a second-time jackpot win by 2034 is over 57%. Even if the average number of jackpot winners per draw drops as low as 1, the chance of a second-time win is still almost 50%.

The desire of people to throw money away doesn't seem likely to disappear anytime soon so I imagine the lottery will still be going in 2034.

Players' choices of numbers will only really be independent if they use the Lucky Dip option but many players do and it seems unlikely that their choices are sufficiently correlated to really make a big difference.

#### What does this all mean?

Well, the headlines that will inevitably accompany any two-time jackpot winner will probably be along the lines of "Is this the luckiest man/woman in Britain?"

Although this will be true (in some sense), the idea that the chance of this happening is 1 in 200,000,000,000,000 (two hundred trillion) is nonsense.

In fact we should really be more surprised if it never happens!

John Talbot