The hypergeometric distribution models the number of successes in a sample drawn from a finite population without replacement. It is useful when each draw changes the population, such as selecting cards from a deck, testing parts from a shipment, or choosing students from a class. The key idea is that the probability of success is not constant from draw to draw.
This makes it different from many simpler probability models used for repeated independent trials.
A hypergeometric problem has a population size N, a number of successes in the population K, a sample size n, and a random variable X for the number of successes in the sample. Its probability formula counts the number of ways to choose x successes and n - x failures, then divides by the number of ways to choose any n items from the population. The distribution is closely related to combinations because order does not matter.
When the population is very large compared with the sample, the binomial distribution can sometimes be used as an approximation.
Key Facts
- Hypergeometric distribution applies when sampling is done without replacement from a finite population.
- X = number of successes in the sample.
- P(X = x) = [C(K, x) C(N - K, n - x)] / C(N, n).
- N = population size, K = number of successes in the population, n = sample size, x = observed successes.
- Mean: E(X) = nK/N.
- Variance: Var(X) = n(K/N)(1 - K/N)((N - n)/(N - 1)).
Vocabulary
- Hypergeometric distribution
- A probability distribution for the number of successes in a fixed size sample drawn without replacement from a finite population.
- Without replacement
- A sampling method where an item is not returned to the population after it is selected.
- Success
- An item or outcome that has the characteristic being counted in the problem.
- Combination
- A count of ways to choose items when the order of selection does not matter.
- Binomial distribution
- A probability distribution for the number of successes in independent trials with the same probability of success each time.
Common Mistakes to Avoid
- Using the binomial formula automatically is wrong because hypergeometric draws are not independent when sampling is without replacement.
- Treating the probability of success as constant is wrong because each draw changes the remaining population unless replacement occurs.
- Forgetting the failure term C(N - K, n - x) is wrong because the sample must include both the chosen successes and the chosen failures.
- Allowing impossible values of x is wrong because x cannot exceed K, cannot exceed n, and must leave enough failures to complete the sample.
Practice Questions
- 1 A box contains 12 red marbles and 8 blue marbles. If 5 marbles are drawn without replacement, what is the probability of drawing exactly 3 red marbles?
- 2 A shipment has 50 parts, including 6 defective parts. An inspector randomly selects 10 parts without replacement. Find the expected number of defective parts in the sample.
- 3 Explain why drawing 4 cards from a standard deck without replacement is modeled by a hypergeometric distribution if X counts the number of aces, and explain why a binomial model would not be exact.