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.
Understanding Statistics: The Hypergeometric Distribution
The counting idea is easier to understand by focusing on complete samples rather than the order of individual draws. Suppose a warehouse contains items of two types, acceptable items and defective items. A sample with a certain number of defects can be formed in many different ways.
First count the ways to select the required defective items from all defective items. Then count the ways to fill the remaining sample positions with acceptable items. Multiplying these counts gives the number of samples with that result.
Every possible sample of the required size belongs in the total count. This approach works because a sample is a group. Selecting item A then item B gives the same sample as selecting item B then item A.
Not every value for the number of successes is possible. A sample cannot contain more successes than exist in the whole population. It cannot contain more successes than the sample size.
There may even be a minimum number of successes required. For example, if a group has twelve members, five with a particular trait, and a sample contains ten members, the sample must include at least three members with that trait.
There are only seven members without it, so ten selected members cannot all come from that group. Checking these limits before calculating a probability prevents impossible answers and reduces mistakes with combinations.
The average number of successes has a simple interpretation. It equals the sample size multiplied by the proportion of successes in the population. This is a long run average, not a promise for one sample.
The spread around that average is smaller than it would be if each draw were independent. Once a success is selected, one fewer success remains for later selection. This creates a negative connection between draws.
A sample that gets unusually many successes early is slightly less likely to get more later. The variance includes a finite population correction that measures this effect.
When the sample is a large part of the population, the correction strongly reduces the spread. When the sample is tiny compared with the population, the correction is close to one.
Students meet this model in quality control, wildlife studies, school surveys, and audits. An inspector may choose products from a small shipment. A biologist may tag animals, then study a fixed group from a pond.
A teacher may select papers for a check from one class. In each case, the model only fits if the selection is genuinely random and each item can be classified clearly as a success or failure. Bias in the selection process cannot be repaired by a probability formula.
When solving problems, identify the full population first, decide what counts as a success, state the sample size, and check whether selected items return to the population. That final detail determines whether this distribution is appropriate.
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.