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March 27

Lottery probability

Hi

let's say in a lottery game one has to choose k numbers out of N. In such situation, it's obvious the probability to choose exactly the right k numbers is 1/C(N,k). But How do I find the chanaces to choose correctly just j (j<=k) of the k numbers (I am looking for the derivation of the answer)? Thanks, 212.179.61.122 (talk) 09:15, 27 March 2014 (UTC)[reply]

To be sure I understand: you choose k numbers, the lottery draws k numbers, and you win if you have at least j in common? In that case, let's start by finding the probability of having exactly j in common, which we'll do by counting the number of ways to have j in common. First, I select the common j from the k which were drawn: C(k, j). Then I fill in the remaining k - j from the undrawn N-k: C(N-k, k-j). So

{\frac {C(k,j)\cdot C(N-k,k-j)}{C(N,k)}}

is the probability of getting exactly j in common. If you want at least j in common, then sum from j to k:

\sum _{i=j}^{k}{\frac {C(k,i)\cdot C(N-k,k-i)}{C(N,k)}}

.--80.109.80.78 (talk) 09:59, 27 March 2014 (UTC)[reply]

Thank you very much. Is it actually the hypergeometric distribution? 212.179.46.21 (talk) 11:22, 27 March 2014 (UTC)[reply]

Well, it's a special case. Using the notation of that page, you have K = n.--149.148.255.195 (talk) 13:54, 27 March 2014 (UTC)[reply]