The Annals of Applied Probability

How Many IID Samples Does it Take to See all the Balls in a Box?

Thomas M. Sellke

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Suppose a box contains $m$ balls, numbered from 1 to $m$. A random number of balls are drawn from the box, their numbers are noted and the balls are then returned to the box. This is done repeatedly, with the sample sizes being iid. Let $X$ be the number of samples needed to see all the balls. This paper uses Markov-chain coupling to derive a simple but typically very accurate approximation for $EX$ in terms of the sample size distribution. The approximation formula generalizes the formula found by Polya for the special case of fixed sample sizes.

Article information

Ann. Appl. Probab., Volume 5, Number 1 (1995), 294-309.

First available in Project Euclid: 19 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Coupon collector's problem Markov chains Markov-chain coupling


Sellke, Thomas M. How Many IID Samples Does it Take to See all the Balls in a Box?. Ann. Appl. Probab. 5 (1995), no. 1, 294--309. doi:10.1214/aoap/1177004841.

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