Description: The hypergeometric distribution is a discrete probability distribution that describes the probability of obtaining a specific number of successes in a given number of extractions from a finite population, without replacement. Unlike the binomial distribution, where extractions are made with replacement, the hypergeometric distribution applies in situations where the population size is fixed and extractions affect the composition of the population. This distribution is characterized by three parameters: the total population size (N), the number of successes in the population (K), and the number of extractions performed (n). The probability function of the hypergeometric distribution allows for the calculation of the probability of obtaining exactly k successes in n extractions, which is useful in various statistical applications. The formula for calculating this probability involves combinations, reflecting the discrete nature of the distribution. The hypergeometric distribution is particularly relevant in contexts where one wishes to understand the variability of results in samples drawn from limited populations, such as in quality studies, sampling in biological populations, or in decision-making situations where specific groups within a larger set are evaluated.
History: The hypergeometric distribution was formally described in the 19th century, although its foundations trace back to earlier works in probability theory. One of the first to study such problems was the French mathematician Pierre-Simon Laplace, who significantly contributed to the theory of probabilities in general. However, it was in the context of modern statistics that the hypergeometric distribution began to be used more systematically, especially in the field of sampling. Over time, it has been integrated into various areas of research and practical application, establishing itself as an essential tool in inferential statistics.
Uses: The hypergeometric distribution is used in various fields, including biology, industrial quality control, and social research. It is particularly useful in sampling situations where elements are drawn from a finite population without replacement. For example, it can be applied in genetic studies to determine the probability of obtaining a specific number of individuals with a particular trait in a sample drawn from a larger population. It is also used in quality control to assess the probability of finding a certain number of defective products in a sample from a batch.
Examples: A practical example of the hypergeometric distribution is sampling from an urn containing 10 red balls and 5 blue balls. If 5 balls are drawn without replacement, the hypergeometric distribution can be used to calculate the probability that exactly 3 of the drawn balls are red. Another case is in medical research, where one can assess the probability that a specific number of patients respond to a treatment in a limited study group.
