Expected number of blue balls drawn from a sack of m red balls and n blue balls?

[Somefantastik]

Can someone help me break this down?

$$\Sigma^{k}_{i=1}\frac{i \left(^{n}_{i}\right)\left(^{m}_{k-i}\right)}{\left(^{m+n}_{k}\right)}$$

 

[Roberto Bramb]

k*n/(m+n)

 

[Somefantastik]

Thanks for your help, but I had the answer and was really looking for the process.

 

[ssd]

Look at the expectation of a Hypergeometric variable.

 

[Adeimantus]

Can someone help me break this down?

$$\Sigma^{k}_{i=1}\frac{i \left(^{n}_{i}\right)\left(^{m}_{k-i}\right)}{\left(^{m+n}_{k}\right)}$$

First translate from math to English: there are m red balls and n blue balls in a sack from which you randomly draw k balls. What is the expected number of blue balls drawn? Now translate back into math: try using indicator random variables $X_{j}$ which equal 1 if the j-th drawn ball is blue and 0 if it is red. Now define the random variable

$$X = \sum_{j=1}^{k} X_{j}$$

and compute the expected value of that and hopefully you'll get the answer that Roberto gave.


addendum: doh! After all that I just realized you can factor the answer out of the sum. Then use the illustration of selecting balls to see what the resulting sum must be.