Calculator Input
Example Data Table
| Population N | Successes K | Sample n | Request | Probability |
|---|---|---|---|---|
| 20 | 5 | 4 | P(X = 1) | 0.469556 |
| 50 | 12 | 6 | P(X ≥ 2) | 0.447225 |
| 60 | 15 | 8 | P(X ≤ 1) | 0.350285 |
| 100 | 18 | 12 | P(1 ≤ X ≤ 3) | 0.779198 |
Formula Used
Hypergeometric probability: P(X = x) = [C(K, x) × C(N − K, n − x)] / C(N, n)
Expected value: E(X) = n × K / N
Variance: Var(X) = n × (K / N) × (1 − K / N) × [(N − n) / (N − 1)]
Here, N is population size, K is the number of successes, n is sample size, and x is observed successes in the sample.
How to Use This Calculator
Enter the total population size first.
Enter how many population items count as successes.
Enter the sample size drawn without replacement.
Select the probability mode that matches your question.
Enter one success count, or two counts for a range.
Choose decimal places for the displayed output.
Press the calculate button to see the result above the form.
Use the CSV or PDF buttons to keep a copy.
Probability of Random Sample Overview
Probability of a random sample matters in surveys, audits, genetics, and quality control. This calculator estimates the chance of observing selected success counts in a sample drawn from a finite population. It uses exact hypergeometric probability. That makes it useful when sampling happens without replacement. You can test one count, cumulative tails, or a closed interval. The output also shows expected successes, variance, standard deviation, and the full distribution table for feasible results.
Why Finite Population Sampling Changes the Result
Random sampling without replacement changes probabilities after every draw. Each selected item affects what remains in the population. Because of that, the binomial model is not always appropriate. The hypergeometric model is usually the correct choice for finite sets. Examples include defective products in a batch, approved files in a folder, or marked cards in a deck. This calculator helps you measure those outcomes quickly and clearly.
Inputs That Control Sample Probability
The most important inputs are population size, success states, and sample size. Population size is the total number of items. Success states are items with the target characteristic. Sample size is the number drawn. After entering those values, choose the probability type. You can find the chance of exactly x successes, at least x successes, at most x successes, or between two values. This creates a flexible tool for classroom practice and real data work.
How to Read the Output
Interpret the result in context. A high exact probability means that count is common under the stated conditions. A small tail probability can signal an unusual sample outcome. The expected value gives the long run average number of successes across repeated samples. The variance and standard deviation show how much counts can spread around that average. Use the example table, formula section, and exports to document your work with confidence.
Why the Table and Exports Help
This page also supports careful checking. The feasible sample success range is displayed automatically, so impossible values become easy to spot. That helps prevent input mistakes during homework, reporting, and statistical review. Export options simplify sharing with teammates, teachers, or clients. When you need transparent reasoning, the distribution table makes every possible success count visible. That improves interpretation, supports verification, and builds intuition about finite population sampling behavior. Practical output supports checking and repeated decisions.
Frequently Asked Questions
1. What distribution does this calculator use?
It uses the hypergeometric distribution. That model fits random sampling without replacement from a finite population.
2. When should I use this calculator?
Use it when you draw items from a known population and each draw changes the remaining composition. Audits, card draws, inspections, and surveys are common examples.
3. What does exact probability mean?
Exact probability means the chance of observing one specific number of successes in the sample, such as exactly 2 defective items.
4. What is the difference between at least and at most?
At least adds probabilities from your chosen count upward. At most adds probabilities from the lowest feasible count up to your chosen count.
5. Can I use this for sampling with replacement?
No. Sampling with replacement is usually modeled by the binomial distribution because the success probability stays constant across draws.
6. Why can the calculator return zero probability?
Zero appears when the requested success count is impossible. For example, you cannot observe more successes than the sample size or available success states.
7. What does expected successes tell me?
Expected successes is the long run average count you would see if the same sampling process were repeated many times.
8. Does a bigger sample always increase exact probability?
No. A larger sample changes the whole distribution. Some exact counts become less likely, while cumulative probabilities may rise or fall.