Hypergeometric Distribution

Definition

The hypergeometric distribution is a discrete probability distribution that describes the probability of obtaining a specific number of successes in a sequence of draws from a finite population without replacement. It is particularly useful in scenarios where you are sampling from a small group and the order of selection does not matter, connecting to concepts like discrete outcomes and probability calculations in a defined set.

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. Unlike the binomial distribution, which assumes independence and replacement, the hypergeometric distribution accounts for the changing probabilities as items are drawn without replacement.
3. The probability mass function for the hypergeometric distribution can be expressed as: $$P(X = k) = \frac{{\binom{K}{k} \binom{N-K}{n-k}}}{{\binom{N}{n}}}$$ where X is the number of successes in the sample.
4. The expected value (mean) of a hypergeometric distribution can be calculated using the formula: $$E(X) = n \cdot \frac{K}{N}$$ which shows how many successes to expect based on proportions.
5. Hypergeometric distributions are commonly used in quality control and ecological studies where sampling without replacement is typical.

Review Questions

• How does the hypergeometric distribution differ from the binomial distribution in terms of sampling methods?
  • The key difference between the hypergeometric distribution and the binomial distribution lies in how samples are taken. The hypergeometric distribution is used for sampling without replacement, meaning once an item is selected, it cannot be chosen again, which changes the probabilities for subsequent selections. In contrast, the binomial distribution assumes independence by allowing for replacement, thus keeping the probabilities constant throughout the trials.
• Discuss the significance of the parameters N, K, and n in determining the probabilities within a hypergeometric distribution.
  • In a hypergeometric distribution, N represents the total population size, K denotes the number of successful outcomes in that population, and n is the number of draws being taken. These parameters are crucial because they define the context of the sampling process. The values of K and N directly influence how many successes one might expect in n draws, while n determines how many opportunities there are to achieve success. This relationship highlights how finite populations and specific outcomes interact under this distribution model.
• Evaluate a scenario where using a hypergeometric distribution would be more appropriate than using a binomial distribution and explain your reasoning.
  • Consider a quality control situation in a factory where there are 100 light bulbs, 10 of which are defective. If an inspector randomly selects 5 light bulbs without replacement to test them for defects, using a hypergeometric distribution is more appropriate than a binomial distribution. This is because once a bulb is chosen and tested, it cannot be selected again, thereby affecting the remaining probabilities of selecting defective bulbs. The hypergeometric model accurately reflects this dependency among selections, making it ideal for analyzing this situation.

Related terms

Binomial Distribution: A discrete probability distribution that represents the number of successes in a fixed number of independent Bernoulli trials, with each trial having the same probability of success.

Population: The entire group from which a sample is drawn for statistical analysis, which can be finite or infinite.

Sampling Without Replacement: A sampling method where each member of the population can only be chosen once, affecting the probabilities of subsequent draws.