There are n balls in a pool. Pick up one ball each time and then put it back to the pool. Do it k times, where k > n. What’s the probability that all balls have been picked up?
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Do you have some thoughts on how to tackle this problem? – hardmath Jan 10 '16 at 14:36
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2This is often referred to as the coupon collector's problem. – Marcus M Jan 10 '16 at 14:37
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@MarcusM Cool, thx! Is there any book collecting most popular problems on probability theory? – lu yuan Jan 10 '16 at 14:54
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@hardmath thx, I will check it – lu yuan Jan 10 '16 at 14:55
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Sorry, I've changed my mind. The other problem is sampling without replacement, while your problem is sampling with replacement. – hardmath Jan 10 '16 at 15:01
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@MarcusM it seems give the expectation and probability bound, but not the exactly probability function. – lu yuan Jan 10 '16 at 15:12
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Related to some of the questions linked to http://math.stackexchange.com/questions/26772/birthday-coverage-problem – Henry Jan 10 '16 at 15:19
1 Answers
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This is an occupancy problem.
You can find an expression by inclusion-exclusion, but a simpler expression comes from Stirling numbers of the second kind, and the probability is $$\dfrac{S_2(k,n) \,n!}{n^k}$$ where the numeration is the number of equally probable draw patterns where each ball is picked at least once while the denominator is the total possible number of equally probable draw patterns.
Henry
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