# Lottery probability

In the UK the lottery uses numbers 1 to 49 and a total of six numbers are picked. It has been said may times that there is as much chance of numbers 1 2 3 4 5 6 to be picked as any other random combination.

My question is this:

Let's say that the first 3 numbers to come out are 1,2 and 3. What are the chances of a number between 1 and 10 coming out next Vs a number between 11 and 20?

There are obviously less numbers between 1 and 10 now that we already lost 1 to 3, so surely the probability is that a number between 11 and 20 is more likely? In which case, the chances of a lottery selection of 1,2,3,4,5,6 is less likely than 2,12,21,28,32,47 for example...

-
That is true, but a priori to the first three drawings the probability is the same. Once you drew three balls it becomes a conditional probability. –  Asaf Karagila Oct 9 '10 at 14:10
If you consider 'numbers with digit 2 in them', by your logic, 1,2,3,4,5,6 has higher probability than 2,12,21,28,32,42... –  Aryabhata Oct 9 '10 at 14:28

The chances of the next number being between 1 and 10 is $\frac{7}{49}$, as opposed to the probability of it being between 11 and 20 being $\frac{10}{49}$. So, among other things, it is less likely that a lottery ticket will have only numbers between 1 and 10, as opposed to numbers between 1 and 20. However, that does not mean that a given ticket with numbers between 1 and 10 is less likely then a given ticket with numbers between 1 and 20. The fact that there are more tickets with numbers between 1 and 20 exactly cancels with the higher chance of getting such a ticket.