Hypergeometric Probability

Definition

Hypergeometric probability is a discrete probability distribution that describes the probability of a certain number of successes in a fixed number of trials, without replacement, from a finite population. It is particularly useful in situations where the population size is relatively small, and the probability of success in each trial is not constant across trials.

5 Must Know Facts For Your Next Test

1. Hypergeometric probability is used when sampling without replacement from a finite population, unlike the Binomial distribution which assumes sampling with replacement.
2. The hypergeometric probability formula takes into account the size of the population, the number of successes in the population, and the number of trials or samples taken.
3. Hypergeometric probability is often used in quality control, survey sampling, and genetics to model the probability of obtaining a certain number of successes in a fixed number of trials.
4. The hypergeometric distribution is related to the Binomial distribution, but differs in that the probability of success changes with each trial due to the sampling without replacement.
5. Hypergeometric probability is particularly useful when the population size is relatively small, and the probability of success in each trial is not constant across trials.

Review Questions

• Explain how hypergeometric probability differs from the Binomial distribution.
  • The key difference between hypergeometric probability and the Binomial distribution is that hypergeometric probability models sampling without replacement from a finite population, whereas the Binomial distribution assumes sampling with replacement from an infinite population. This means that in the hypergeometric case, the probability of success changes with each trial as the population size decreases, whereas in the Binomial case, the probability of success remains constant across trials.
• Describe a real-world scenario where hypergeometric probability would be the appropriate probability model to use.
  • One example where hypergeometric probability would be the appropriate model is in quality control testing of a manufactured product. Suppose a manufacturer produces a batch of 1,000 items, and they want to test a sample of 50 items for defects. The hypergeometric probability distribution would be the appropriate model to use, as the population size is finite (1,000 items), the sampling is without replacement (the tested items are removed from the population), and the probability of success (finding a defect) is not constant across trials due to the decreasing population size.
• Explain how the hypergeometric probability formula takes into account the size of the population, the number of successes in the population, and the number of trials or samples taken.
  • $$P(X = x) = \frac{\binom{M}{x}\binom{N-M}{n-x}}{\binom{N}{n}}$$ Where: - $N$ is the total population size - $M$ is the number of successes (or desired outcomes) in the population - $n$ is the number of trials or samples taken - $x$ is the number of successes (or desired outcomes) in the $n$ trials The formula accounts for the population size, the number of successes in the population, and the number of trials by calculating the probability of obtaining $x$ successes out of $n$ trials, given the finite population size and the number of successes in the population.

Related terms

Probability Distribution: A probability distribution is a mathematical function that describes the possible values and their corresponding probabilities of a random variable.

Discrete Probability Distribution: A discrete probability distribution is a probability distribution where the random variable can only take on a countable number of distinct values.

Finite Population: A finite population is a population that has a known, fixed, and limited number of elements or members.