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Finding cardinality of set based on random selection from set

• To: mathgroup at smc.vnet.net
• Subject: [mg75390] Finding cardinality of set based on random selection from set
• From: "Kelly Jones" <kelly.terry.jones at gmail.com>
• Date: Sat, 28 Apr 2007 05:53:16 -0400 (EDT)
• References: <26face530704242008sc104934o891814d2be51bcf1@mail.gmail.com>

How do I use Mathematica to solve this problem?

 I selected with replacement 100 times from a set of N objects. The
 results:

 1 item came up 4 times
 1 item came up 3 times
 10 items came up 2 times each
 73 items came up 1 time each
 the other N-85 items did not come up at all

 Based on this histogram, what's the 1) most likely value of N, and 2)
 the probability distribution of N?

 Of course, I'm actually interested in the general solution: if I
 select with replacement M times from a set of N objects, and get a
 resulting histogram H, what's the probability distribution of N with
 respect to M and H?