Hypergeometric Distribution

Definition

The hypergeometric distribution is a discrete probability distribution that describes the probability of a certain number of successes in a fixed number of draws, without replacement, from a finite population. It is used to model situations where a sample is drawn from a population without replacement, and the interest lies in the number of items with a particular characteristic in the sample.

5 Must Know Facts For Your Next Test

1. The hypergeometric distribution is used when the sample size is small compared to the population size, and the population size is finite.
2. The hypergeometric distribution is characterized by three parameters: the population size, the number of items with a particular characteristic in the population, and the sample size.
3. The hypergeometric distribution is often used in quality control, where the goal is to determine the probability of finding a certain number of defective items in a sample drawn from a larger population.
4. The hypergeometric distribution is more appropriate than the binomial distribution when the population size is small and the sample size is large relative to the population size.
5. The hypergeometric distribution can be used to model a variety of real-world situations, such as drawing balls from an urn, selecting cards from a deck, or inspecting a batch of manufactured items.

Review Questions

• Explain the key differences between the hypergeometric distribution and the binomial distribution.
  • The main difference between the hypergeometric distribution and the binomial distribution is that the hypergeometric distribution models sampling without replacement from a finite population, while the binomial distribution models independent trials with two possible outcomes (success or failure) and replacement. In the hypergeometric distribution, the probability of success on each draw depends on the number of items with the desired characteristic remaining in the population, whereas in the binomial distribution, the probability of success on each trial is constant. Additionally, the hypergeometric distribution is more appropriate when the sample size is large relative to the population size, while the binomial distribution is more suitable when the sample size is small compared to the population size.
• Describe the three key parameters that characterize the hypergeometric distribution and explain how they influence the probability of success.
  • The three parameters that characterize the hypergeometric distribution are: the population size (N), the number of items with a particular characteristic in the population (K), and the sample size (n). The population size (N) represents the total number of items in the population, the number of items with a particular characteristic (K) determines the probability of selecting a successful item, and the sample size (n) represents the number of items drawn from the population. The probability of success in the hypergeometric distribution is influenced by the relationship between these three parameters, as the probability of selecting a successful item decreases with each draw due to the sampling without replacement.
• Explain how the hypergeometric distribution can be applied to real-world situations and discuss the advantages of using this distribution over other probability distributions.
  • The hypergeometric distribution can be applied to a variety of real-world situations where a sample is drawn from a finite population without replacement, and the interest lies in the number of items with a particular characteristic in the sample. Examples include quality control in manufacturing, where the goal is to determine the probability of finding a certain number of defective items in a sample, or drawing balls from an urn without replacement. The hypergeometric distribution is advantageous over other probability distributions, such as the binomial distribution, when the population size is small and the sample size is large relative to the population size. In these cases, the hypergeometric distribution provides a more accurate representation of the probability of success, as it takes into account the finite nature of the population and the sampling without replacement. This makes the hypergeometric distribution a valuable tool for decision-making in various fields, including quality control, epidemiology, and sampling theory.

Related terms

Binomial Distribution: A discrete probability distribution that describes the probability of a certain number of successes in a fixed number of independent trials, where each trial has only two possible outcomes (success or failure).

Sampling without Replacement: A sampling method where items are drawn from a population, and once an item is selected, it is not returned to the population before the next draw.

Finite Population: A population with a fixed, known size, as opposed to an infinite population.