Hypergeometric Distribution Calculator: Without Replacement Probability
Use this hypergeometric distribution calculator to find exact, cumulative, at least, at most, and between probabilities for sampling without replacement. It helps with HYPGEOM.DIST-style checks, card problems, quality-control samples, audit samples, finite populations, and statistics homework.
- Without replacement Use the right probability model when items are sampled from a finite population without replacement.
- Exact probability Calculate P(X = k) for a population size, success count, sample size, and observed successes.
- Cumulative tails Find P(X <= k), P(X < k), P(X >= k), or P(X > k) for sampling problems.
- Range probability Calculate the probability that the number of successes falls between two counts.
Calculate Hypergeometric Probability
Enter the population size, how many successes are in that population, how many items are sampled, and the success count you care about. The calculator assumes random sampling without replacement, so the probability of success changes after each draw.
Inputs
Results
For a population of 52 with 13 successes and 5 draws, the probability of exactly 2 successes is about 27.43%.
The highlighted bar shows k = 2.
| Formula | P(X = 2) = C(13, 2) * C(39, 3) / C(52, 5) |
|---|---|
| Probability | P(X = 2) = 0.2743 |
| Support | Possible success counts: 0 to 5. |
Hypergeometric Probability Table
The table shows exact probabilities and cumulative probabilities for the current population and sample values.
| Successes k | P(X = k) | P(X <= k) | P(X >= k) |
|---|
Hypergeometric Distribution Formula
The hypergeometric formula counts how many ways the sample can contain k successes and n minus k failures, divided by the number of possible samples.
| Exact probability | P(X = k) = C(K, k) * C(N - K, n - k) / C(N, n) |
|---|---|
| Mean | mean = n * K / N |
| Variance | variance = n * (K / N) * (1 - K / N) * ((N - n) / (N - 1)) |
| Support | possible k values run from max(0, n + K - N) to min(K, n) |
Common Mistakes
- Using binomial probability when the sample is drawn without replacement from a finite population.
- Making the population success count larger than the total population size.
- Using a sample size larger than the population size.
- Forgetting that some success counts are impossible because of the population and sample limits.
How to Calculate Hypergeometric Probability Step by Step
Hypergeometric homework is mostly about recognizing that the sample is taken without replacement and matching each number to the right symbol.
- Identify the population size N.
- Identify the number of successes in the population, K.
- Identify the sample size, n.
- Identify the success count k or range of success counts in the question.
- Use exact probability for wording like exactly k successes.
- Use cumulative probability for wording like at most, fewer than, at least, or more than.
- Check the possible support because some k values are impossible in finite population problems.
Hypergeometric Probability Examples
These examples are useful for checking HYPGEOM.DIST-style output, card-draw questions, and finite-population sampling homework.
| Question type | Inputs | Result | Interpretation |
|---|---|---|---|
| Exactly 2 successes | N = 52, K = 13, n = 5, k = 2 | P(X = 2) = 0.2743 | About 27.43% of repeated samples would contain exactly 2 successes. |
| At most 2 successes | N = 52, K = 13, n = 5, k = 2 | P(X <= 2) = 0.9072 | About 90.72% of repeated samples would contain 2 or fewer successes. |
| At least 2 successes | N = 52, K = 13, n = 5, k = 2 | P(X >= 2) = 0.3670 | About 36.70% of repeated samples would contain 2 or more successes. |
| Between 1 and 3 successes | N = 52, K = 13, n = 5, lower = 1, upper = 3 | P(1 <= X <= 3) = 0.7672 | About 76.72% of repeated samples would land in this success-count range. |
When This Calculator Helps
Statistics homework
Use it when your problem includes a finite population, a known number of success items, and a fixed sample taken without replacement.
Real-world samples
Use it for card draws, audit sampling, quality checks, inspection lots, raffle-style selections, and survey samples from a limited list.
Hypergeometric Distribution FAQs
When should I use the hypergeometric distribution?
Use it when you draw a fixed-size sample without replacement from a finite population that contains successes and failures.
How is hypergeometric different from binomial?
Binomial probability assumes independent trials with the same success probability. Hypergeometric probability is for sampling without replacement, so the success probability changes after each draw.
What do N, K, n, and k mean?
N is the total population size, K is the number of successes in the population, n is the sample size, and k is the observed number of successes in the sample.
Can the calculator show impossible values?
Yes. If the chosen success count is outside the possible support, the exact probability is 0 and the support row shows the feasible range.
What does sampling without replacement mean?
Sampling without replacement means each selected item is removed from the population before the next draw, so the success probability changes after each draw.
Is this like Excel HYPGEOM.DIST?
Yes. It follows the same hypergeometric probability ideas used in Excel-style HYPGEOM.DIST work for exact and cumulative probabilities. Small differences can appear from rounding.
When should I use binomial instead?
Use binomial when trials are independent and the success probability stays the same. Use hypergeometric when you sample from a finite population without replacement.
Related Statistics Resources
If the problem moves from probability to testing, assumptions, or written interpretation, these pages are good next steps.
Need Help With a Hypergeometric Probability Problem?
Statskan can help you choose the right probability model, calculate the result, show the formula, and write the interpretation clearly for your statistics homework.