Calculator inputs
Use without replacement for a classic simple random sample from a finite population. Use with replacement to compare independent repeated draws.
Formula used
1) Without replacement: Hypergeometric model
P(X = x) = [C(K, x) × C(N - K, n - x)] / C(N, n)
Use this when a sample is drawn from a finite population and items are not returned before the next draw.
Expected successes: E(X) = n × (K / N)
Variance: Var(X) = n × (K / N) × (1 - K / N) × [(N - n) / (N - 1)]
2) With replacement: Binomial model
P(X = x) = C(n, x) × px × (1 - p)n - x
Here, p = K / N. Each draw stays independent because the population composition resets after every selection.
Expected successes: E(X) = n × p
Variance: Var(X) = n × p × (1 - p)
3) Core interpretation
A single randomly sampled observation is a success with probability K / N.
The set of probabilities attached to all possible values of a random variable is called its probability distribution.
How to use this calculator
- Enter the total population size N.
- Enter the number of target items in that population, K.
- Enter the sample size n.
- Choose whether the sampling is without replacement or with replacement.
- Select the probability request type, such as exact, at least, at most, between, or none.
- Enter the relevant success count value.
- Press Calculate probability to see the result above the form.
- Review the table, chart, and export the result to CSV or PDF if needed.
Example data table
| Scenario | N | K | n | Model | Request | Result |
|---|---|---|---|---|---|---|
| Product defect screening | 80 | 12 | 8 | Without replacement | P(X ≥ 1) | 0.744994 (74.50%) |
| Survey response classification | 150 | 45 | 12 | Without replacement | P(X = 4) | 0.240938 (24.09%) |
| Compliance file review | 60 | 9 | 6 | Without replacement | P(1 ≤ X ≤ 2) | 0.601990 (60.20%) |
| Repeated independent draws | 100 | 20 | 10 | With replacement | P(X ≤ 3) | 0.879126 (87.91%) |
Frequently asked questions
1) What does this calculator measure?
It calculates the chance of getting exact, minimum, maximum, zero, or ranged target successes in a simple random sample from a finite population.
2) What is the probability that a single randomly sampled observation have a value above the mean?
There is no universal value. For symmetric continuous distributions, it is 0.5. For skewed or discrete distributions, it depends on the underlying distribution.
3) The set of probabilities associated with the values in a random variable's sample space
That is called a probability distribution. It assigns a probability to every possible value in the random variable’s sample space.
4) When should I use the without replacement option?
Use it when one selected item cannot be selected again in the same sample. This is the standard setup for simple random sampling from a finite list.
5) When should I use the with replacement option?
Use it when each draw is independent and the population composition resets after every selection. It is useful for comparisons and repeated independent trials.
6) Why can the exact probability change in unexpected ways?
Because the sample size, target share, and request type interact. An exact count may become less likely while a cumulative event, like at least one success, becomes more likely.
7) Why is expected value not always the most likely count?
The expected value is a long-run average, not a guarantee. The most likely count is the mode, and it may differ from the mean.
8) Can this calculator help with audits, surveys, and quality checks?
Yes. It is useful when you need the probability of finding target cases, defects, responses, or flagged records in a random sample from a known population.