Hypergeometric Probability Formula

Definition

The hypergeometric probability formula is a discrete probability distribution used to calculate the probability of obtaining a certain number of successes in a fixed number of trials, without replacement, from a finite population. It is particularly useful in scenarios where a sample is drawn from a population without replacement, such as quality control testing or sampling from a limited resource pool.

5 Must Know Facts For Your Next Test

1. The hypergeometric probability formula is used when the population size, number of successes in the population, and sample size are known.
2. The formula calculates the probability of obtaining a specific number of successes in a sample drawn from the population without replacement.
3. The hypergeometric distribution is a discrete probability distribution, meaning it can only take on integer values and the probabilities must sum to 1.
4. The hypergeometric distribution is often used in quality control, where a sample is tested for defects in a finite population of items.
5. The hypergeometric distribution is also used in sampling without replacement, such as in polling or market research, where the population size is limited.

Review Questions

• Explain the key assumptions and requirements for using the hypergeometric probability formula.
  • The key assumptions and requirements for using the hypergeometric probability formula are: 1) the population size is finite and known, 2) the sample is drawn without replacement, meaning items are not returned to the population after being selected, and 3) the number of successes (e.g., defective items) in the population is known. These assumptions ensure that the probability of obtaining a certain number of successes in the sample is accurately represented by the hypergeometric distribution.
• Describe how the hypergeometric probability formula differs from the binomial probability formula.
  • The key difference between the hypergeometric probability formula and the binomial probability formula is that the hypergeometric formula accounts for sampling without replacement from a finite population, whereas the binomial formula assumes sampling with replacement from an infinite population. In the hypergeometric case, the probability of success on each trial depends on the number of successes remaining in the population, which decreases with each draw. This makes the hypergeometric formula more appropriate for situations where the population size is limited, such as in quality control testing or market research sampling.
• Analyze how the parameters of the hypergeometric probability formula (population size, number of successes in the population, and sample size) influence the calculated probabilities.
  • The parameters of the hypergeometric probability formula have a significant impact on the calculated probabilities. As the population size increases, the hypergeometric probabilities approach the binomial probabilities, since the impact of sampling without replacement becomes negligible. Increasing the number of successes in the population generally increases the probability of obtaining more successes in the sample, while increasing the sample size increases the probability of obtaining a larger number of successes. Understanding how these parameters influence the hypergeometric probabilities is crucial for correctly interpreting and applying the formula in real-world scenarios.

Related terms

Discrete Probability Distribution:

A probability distribution that describes the probability of a random variable taking on a finite or countable number of possible values.

Sampling without Replacement: A sampling method where items are drawn from a population without being returned, meaning the population size decreases with each draw.

Finite Population:

A population with a known, fixed, and limited number of elements or items.