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Suppose we have $k$ buckets, with an infinite number of balls in each bucket. The balls within one bucket are indistinguishable, those between buckets are. We assign to bucket $i$ a probability $p_i$ (so that $\sum_{i=1}^k p_i=1$). If we draw a ball from one of these buckets with probability $p_i$ without replacing it, and repeat this $N$ times, we are sampling from a multinomial distribution with $N$ trials and $k$ categories. The fact that there is an infinite number of balls in each bucket makes replacement superfluous.

Now, what happens if bucket $i$ only contains $c_i$ balls? The $c_i$ are defined arbitrarily, but such that $\sum_{i=1}^k c_i\ge N$. When drawing from the buckets with probability $p_i$, the number of balls in the bucket has no influence on the outcome as long as there is a ball in the bucket. That is, if a bucket $i$ is depleted, you cannot draw from that bucket anymore. You therefore have to draw from another bucket, with probability $p_j/(1-\sum_{i\in depleted} p_i)$. What is this distribution? Can its pmf be given? Can it be related to the multivariate hypergeometric distribution? Are there approximations for it?

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  • $\begingroup$ It's impossible to tell what you are asking because you do not specify what numbers might be written on the balls in those buckets. In addition, your process is not completely defined, because even when $\sum c_i \ge N$, it will be possible to exhaust one of the buckets during the trials if it has fewer than $N$ balls in it. Did you perhaps mean to stipulate that all the $c_i\ge N$? $\endgroup$ – whuber Dec 18 '14 at 19:22
  • $\begingroup$ I updated the question to make it clearer $\endgroup$ – yannick Dec 19 '14 at 7:35

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