Lottery Probability: Find 7 Numbers Out of 20

[jk22]

I watch lastly a lottery game : there are 80 numbers, from them 20 are selected, and you can choose for example to find 7 numbers out of the 20.

I'm looking for the probability to find 7 numbers, is it simply C(80,7)/C(80,20) or is it 20!/13!/C(80,20) or C(80,13)/C(80,20) ?

 

[FactChecker]

I can tell that none of those are right. Consider the extreme case where the game show picked 80 (instead of 20) out of 80 . Then C(80,80)=1, so all those formulas would give numbers much larger than 1. They must all be wrong. At the moment, I'm not sure what the correct answer is. What about C(20,7) / C(80,7)? That is the ratio of all your possible picks from the special 20 over all your possible picks from the original 80.

 

[jk22]

I thought a bit about the following : I fix 7 numbers over 80, there remains 13 numbers to choose freely among 73, these are the favorable cases, whereas all the cases are 20 numbers chosen between 80, hence : C(73,13)/C(80,20) ?

 

[FactChecker]

I thought a bit about the following : I fix 7 numbers over 80, there remains 13 numbers to choose freely among 73, these are the favorable cases, whereas all the cases are 20 numbers chosen between 80, hence : C(73,13)/C(80,20) ?

My equation above, C(20,7)/C(80,7) gives exactly the same answer as yours (C(73,13)/C(80,20) = 2.4402556E-5). I'm not sure I follow your reasoning, but since we came at it from two different logic directions, I bet they are both right. I tried to see how they could be identical, but it got too messy for me.

 

[alan2]

Your expressions are the same. The game is Keno. More generally than your case, you choose n numbers from 80 where n is less than or equal to 20. 20 numbers are then drawn and you win if you match k of them where k is less than or equal to n. There are C(80,20) total combinations of 20 drawn from 80. You are holding n numbers. There are C(n,k) combinations of k from n. The rest of the numbers drawn do not match yours, that is 20-k drawn from the remaining 80-n. There are C(80-n,20-k) ways to do that. So the probability of matching k of your n numbers is C(n,k)C(80-n,20-k)/C(80,20). In the special case where you match all of your n numbers, k=n and P=C(80-n,20-n)/C(80,20). If n=7 then P=C(73,13)/C(80,20).