Hi could someone direct me to where I can find a solution for the following problem.
I have a bag of 140 balls they are of 20 colors so each color has 7 balls. I pick 2 balls from the bag. How many different colors can I expect (expected number of colors)
I want to extend this approach to more than 2 picks. So something generic. I know that this is similar to a hypergeometric distributuion but in this case it has more than 2 classes (20 colors)
Just by definition: $$E(X) = 1.P(1) + 2.P(2) = 1.\frac{\binom{20}{1}\binom{7}{2}}{\binom{140}{2}} + 2.\frac{\binom{20}{2}\binom{7}{1}\binom{7}{1}}{\binom{140}{2}}$$ Now suppose $n$ picks: $$E(X) = \sum_{i=1}^{\max(20,n)} i.P(i) = \sum_{i = 1}^{\max(20,n)} i.\binom{20}{i}\prod_{\sum_{k=1}^i j_k = n}\binom{7}{j_k}/\binom{140}{n}$$