0
$\begingroup$

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 ?

$\endgroup$
  • 1
    $\begingroup$ 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
2
$\begingroup$

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

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.