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 ?

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    $\begingroup$ Should each colored ball have $\frac{1-p}{n}$ chance of being picked? $\endgroup$ – CommonerG May 17 '13 at 20:15
  • $\begingroup$ 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. $\endgroup$ – Sid Datta May 17 '13 at 20:26
  • $\begingroup$ I must have read this too quickly. I see now. $\endgroup$ – 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

  • $\begingroup$ I believe the number of picks required to get a colored ball is (1/p), otherwise thanks, this is the answer I am looking for. The final answer is (1/p) * n * Harmonic_sum(n). $\endgroup$ – Sid Datta May 17 '13 at 22:12
  • $\begingroup$ @SidDatta: I got backwards, you are right. Fixed. $\endgroup$ – Ross Millikan May 17 '13 at 22:13

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