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Chapter 5: Problem 70

Explain the hypergeometric probability distribution. Under what conditions is this probability distribution applied to find the probability of a discrete random variable \(x ?\) Give one example of the application of the hypergeometric probability distribution.

Short Answer

Expert verified
The hypergeometric distribution describes the probability of k successes in n trials drawn without replacement, from a population of size N containing K successes. It is applied in scenarios where we have a finite binary population, draw a random sample without replacement, and are interested in the chance of 'success'. An example can be determining the probability of drawing a certain number of red cards from a playing card deck.

Step by step solution

01

Explanation of Hypergeometric Distribution

The hypergeometric distribution is a type of discrete probability distribution that describes the probability of k successes in n draws, without replacement, from a finite population of size N containing K successes. The probability mass function can be expressed as \[P(X=k) = \frac{{C(K, k) * C(N-K, n-k)}}{{C(N, n)}}\]where C is the binomial coefficient.
02

Application Conditions for Hypergeometric Distribution

The hypergeometric distribution can be applied under such conditions: 1) we have a finite population containing two types of items: 'success' and 'failure'; 2) we draw a random sample without replacement; 3) each draw is either a 'success' or 'failure'; 4) we are concerned with the number of 'success' in the sample.
03

Example of Hypergeometric Distribution

Imagine you have a deck of 52 playing cards (this is your population), and you are dealt 5 cards (this is your sample). Of those 52 cards, 26 are red (these are your 'successes'). If you wanted to find out the probability of being dealt exactly 2 red cards, you would use a hypergeometric distribution. This case satisfies the conditions mentioned in Step 2.

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Most popular questions from this chapter

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