Hypergeometric probabilities. An urn contains n balls, out

Chapter , Problem 61

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Hypergeometric probabilities. An urn contains n balls, out of which m are red. We select k of the balls at random. without replacement (Le., selected balls are not put back into the urn before the next selection). What is the probability that i of the selected balls are red? Solution. The sample space consists of the () different ways that we can select k out of the available balls. For the event of interest to occur. we have to select i out of the m red balls, which can be done in (7) ways, and also select k - i out of the n - m balls that are not red, which can be done in ( n k7) ways. Therefore, the desired probability IS for i 0 satisfying i m, i k, and k - i n - m. For all other i, the probability is zero.