The hypergeometric distribution is a discrete probability distribution that describes the likelihood of obtaining a specific number of successes in a sample drawn without replacement from a finite population. It is particularly useful when dealing with scenarios where the population is divided into two distinct categories — such as defective vs. non-defective items or tagged vs. untagged individuals — and where sampling affects the probabilities of subsequent draws. The key parameters of the hypergeometric distribution include the population size (N), the number of success states in the population (K), the number of draws (n), and the number of observed successes (k). The probability of observing exactly k successes in n draws is given by the formula:
P(X=k)=(Kk)(N−Kn−k)(Nn)P(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}}
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Unlike the binomial distribution, which assumes independence between trials due to sampling with replacement (or an infinite population), the hypergeometric distribution accounts for dependence between draws. This makes it highly applicable in quality control, ecological studies, and card games, where each draw changes the composition of the remaining population. As a
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The hypergeometric distribution is a discrete probability distribution that describes the likelihood of obtaining a specific number of successes in a sample drawn without replacement from a finite population. It is particularly useful when dealing with scenarios where the population is divided into two distinct categories — such as defective vs. non-defective items or tagged vs. untagged individuals — and where sampling affects the probabilities of subsequent draws. The key parameters of the hypergeometric distribution include the population size (N), the number of success states in the population (K), the number of draws (n), and the number of observed successes (k). The probability of observing exactly k successes in n draws is given by the formula:
P(X=k)=(Kk)(N−Kn−k)(Nn)P(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}}
more info: https://joinentre.com/feed/635b84e1-35ce-1000-04ca-0a5eafdbe7c0
Unlike the binomial distribution, which assumes independence between trials due to sampling with replacement (or an infinite population), the hypergeometric distribution accounts for dependence between draws. This makes it highly applicable in quality control, ecological studies, and card games, where each draw changes the composition of the remaining population. As a result, the hypergeometric distribution tends to have a slightly different shape and variance than the binomial, particularly when the sample size is a significant fraction of the population. Understanding this distribution helps in making more accurate statistical inferences in finite-sample settings.
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