Hypergeometric Distribution

Definition

The hypergeometric distribution describes the probability of drawing a specific number of successes from a finite population without replacement. This distribution is essential for understanding scenarios where items are selected from a set, such as drawing cards from a deck or selecting individuals from a group, highlighting the relationship between random variables and probability distributions.

5 Must Know Facts For Your Next Test

1. The hypergeometric distribution is defined by three parameters: the population size (N), the number of success states in the population (K), and the number of draws (n).
2. It differs from the binomial distribution because it does not assume that each draw is independent; instead, each draw decreases the population size.
3. The probability mass function (PMF) for the hypergeometric distribution can be expressed as $$P(X = k) = \frac{{\binom{K}{k} \binom{N-K}{n-k}}}{{\binom{N}{n}}}$$, where k is the number of observed successes.
4. This distribution is particularly useful in quality control scenarios, where it helps determine the likelihood of finding defective items in a sample.
5. In applications such as genetics or card games, hypergeometric distribution helps evaluate the probabilities when exact counts of specific items are crucial.

Review Questions

• How does the hypergeometric distribution differ from the binomial distribution in terms of sampling methods?
  • The hypergeometric distribution differs from the binomial distribution mainly in how samples are drawn. While the binomial distribution assumes that each draw is independent and that items are replaced after selection, the hypergeometric distribution deals with samples drawn without replacement. This means that in the hypergeometric case, each selection affects subsequent probabilities since the population size decreases with each draw.
• Discuss how you would apply the hypergeometric distribution to a real-world scenario involving quality control.
  • In quality control, suppose a factory produces 1000 items, 200 of which are defective. If an inspector randomly selects 50 items for testing without replacement, we can use the hypergeometric distribution to calculate the probability of finding a specific number of defective items within that sample. By defining our parameters—total items (N), defective items (K), and sample size (n)—we can use the probability mass function to evaluate different outcomes and make informed decisions about product quality.
• Evaluate how changing one parameter of the hypergeometric distribution affects its probability outcomes, and provide an example.
  • Changing one parameter of the hypergeometric distribution significantly influences its probability outcomes. For instance, if we increase the total population size (N) while keeping K and n constant, the overall probabilities for any given number of successes will generally decrease because there are more non-successes available in the population. For example, if we have a scenario with 10 successes out of 100 total items and we draw 10 without replacement, increasing N to 200 while keeping K at 10 will result in lower chances of drawing multiple successes due to greater competition among non-successes.

Related terms

Binomial Distribution:

A probability distribution that summarizes the likelihood of a given number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success.

Finite Population: A population that consists of a limited number of elements, which is important when considering probabilities and sampling methods.

Sampling Without Replacement: A sampling method where each selection is not replaced back into the population, affecting the probabilities of subsequent selections.