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Chapter 5: Problem 59

Under what conditions would you use the hypergeometric probability distribution to evaluate the probability of \(x\) successes in \(n\) trials?

Short Answer

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Question: Explain the conditions and criteria that must be met in order to use the hypergeometric probability distribution to evaluate the probability of \(x\) successes in \(n\) trials from a finite population. Answer: To use the hypergeometric probability distribution, the following conditions must be met: a finite population of size \(N\) exists with \(M\) success objects and \(N-M\) failure objects; the number of trials (\(n\)) is fixed; trials are performed without replacement; and the probability of success and failure varies across trials. The criteria for evaluating the probability of \(x\) successes in \(n\) trials include: meeting the conditions for using the hypergeometric distribution; ensuring \(n \leq N\); ensuring \(0 \leq x \leq M\); and ensuring \(0 \leq n-x \leq N-M\).

Step by step solution

01

1. Understand hypergeometric distribution

The hypergeometric distribution is a discrete probability distribution that describes the probability of \(x\) successes in \(n\) trials, without replacement, from a finite population of size \(N\), where there are \(M\) success objects and \(N-M\) failure objects.
02

2. Conditions for using the hypergeometric distribution

To use the hypergeometric probability distribution, the following conditions must be met: a) A finite population of size \(N\) exists, containing \(M\) success objects and \(N-M\) failure objects. b) The number of trials (\(n\)) is fixed. c) Each trial is performed without replacement, meaning that once an object is selected, it is not put back into the population. d) The probability of success and failure is not constant across trials, as they depend on the remaining population after each trial.
03

3. Criteria for evaluating the probability of \(x\) successes in \(n\) trials

To evaluate the probability of \(x\) successes in \(n\) trials using the hypergeometric probability distribution, the following criteria must be fulfilled: a) The conditions for using the hypergeometric distribution are met (explained in step 2). b) \(n \leq N\) (the number of trials cannot be greater than the population size). c) \(0 \leq x \leq M\) (the number of successes cannot be negative or greater than the number of success objects in the population). d) \(0 \leq n-x \leq N-M\) (the number of failures cannot be negative or greater than the number of failure objects in the population). Once these conditions and criteria are met, the hypergeometric probability distribution can be used to evaluate the probability of \(x\) successes in \(n\) trials from a population with \(M\) success objects and \(N-M\) failure objects.

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