Drawing balls with replacement, until I have one of each.

A urn has (n+1) types of balls, n of unique colors and the rest black. When picking a ball randomly from the urn, a colored (non black) ball has a probability of p of being picked. Each ball of color has equal probability of being picked, ie each has a (p/n) chance of being picked. The urn has infinite balls/we are picking with replacement.

Let X be the number of picks until we get at least 1 of each n colored balls.

What is the Expected value of X ? What is the probability distribution of X ?

• Should each colored ball have $\frac{1-p}{n}$ chance of being picked? – CommonerG May 17 '13 at 20:15
• Black balls have a (1-p) chance of getting picked, balls of a certain non-black color have a (p/n) chance, since there are n colors. – Sid Datta May 17 '13 at 20:26
• I must have read this too quickly. I see now. – CommonerG May 17 '13 at 20:32

Hint: you have to pick $\frac 1{p}$ balls to get a colored one, so multiply that by the solution to the Coupon collector's problem