# 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. 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... Oct 9 '10 at 14:28
• On a related note, if you want to maximise your expected winnings (and still participate) then you are better served by choosing combinations of numbers unlikely to be chosen by others, so as to minimise your chances of sharing your possible winnings with others. Dec 14 '17 at 5:25

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.