# How long before someone wins the lottery twice?

1. Aug 9, 2011

### Mr.A.Gibson

1. The problem statement, all variables and given/known data
The chance of winning the Uk lottery is 49C6 = 1 in 13,983,816
So the chance of winning the lottery twice with two tickets is 1 in 195,547,109,856

But what is the chance of anyone person winning the lottery twice. Approx 30million tickets are sold each week, average 3 tickets per person. We'll assume that people carry on playing the same way after they've won.

Also, how long will the lottery have to run for before it is more likely that not that someone will have won it twice.
2. Relevant equations

3. The attempt at a solution
Similar method to the number of people in a room for the same birthday I suspect.

30million tickets per week, 30M/13,98M gives approx 2 winners per week

chance of uniqueness
$$\frac{9,999,998}{10,000,000}$$

So the chance of uniqueness after n weeks
$$2\left({\frac{4,999,999!}{(4,999,999-n)!50,000,000^n}}\right)$$
So to work out how long the lottery need to run before its more likely than not that someone will have one twice is:
$$1-2\left({\frac{4,999,999!}{(4,999,999-n)!5,000,000^n}}\right) >0.5$$

Is this method correct and then how could I solve this equation?