Expected Number of Draws without Replacement

Question

An urn contains a white balls, b blue balls and c red balls. Balls are drawn one by one without replacement. Find the expected number of draws needed to get all 3 colors.

Answer 1

You draw a ball first, let's pretend for now that it is $a$. Now you have the urn with two balls problem. Compute the expected number of draws of $b,c$ needed to get both of them. Scale that up by $\frac {a-1+b+c}{b+c}$ to account for the $a$'s you draw along the way. This gives you the expected number of draws if you start with $a$. You can compute the expected number if you start with $b$ and $c$ (or just permute the variables in your solution for $a$), then weight the three values by the fraction in the original bin.