Advanced Sampling Without Replacement Calculator

Sampling Without Replacement Calculator

Use the fields below to evaluate exact, cumulative, and interval probabilities under a hypergeometric model.

Population Size (N)

Total items in the full finite population.

Success States (K)

Items classified as successes before sampling.

Sample Size (n)

Number of draws taken without replacement.

Exact Successes (x)

Exact event for P(X = x).

Range Minimum (a)

Lower bound for interval probability.

Range Maximum (b)

Upper bound for interval probability.

Displayed Decimals

Controls numeric formatting in the results.

Example Data Table

This example models drawing 5 cards from a standard deck and counting aces.

Scenario	Population N	Success States K	Sample Size n	Exact Successes x	Exact Probability	At Least One Success
5 cards drawn, count aces	52	4	5	1	0.299474	0.341158
Interpretation	There is about a 29.9474% chance of drawing exactly one ace and a 34.1158% chance of drawing at least one ace.

Formula Used

Exact probability:

P(X = x) = [C(K, x) × C(N − K, n − x)] / C(N, n)

Expected value and variance:

E[X] = n(K / N)

Var(X) = n(K / N)(1 − K / N)[(N − n) / (N − 1)]

Meaning of symbols:

N = total population size
K = number of success states in the population
n = number of draws without replacement
x = observed number of successes in the sample
C(a, b) = combinations of choosing b from a

This model applies when every sample of size n is equally likely and each draw changes the remaining composition of the population.

How to Use This Calculator

Enter the total population size in N.
Enter how many items count as successes in K.
Enter the number of draws in n.
Choose the exact success count x for the point probability.
Set a and b to measure an interval probability.
Click Calculate Probability to show the results above the form.
Review the graph, summary metrics, and full distribution table.
Use the CSV or PDF buttons to export the output.

FAQs

1. What does sampling without replacement mean?

It means selected items are not returned before the next draw. Each draw changes the remaining population, so probabilities shift from draw to draw.

2. Which distribution does this calculator use?

It uses the hypergeometric distribution. This distribution is appropriate for finite populations, binary classification, and dependent draws made without replacement.

3. When should I use this instead of a binomial calculator?

Use this calculator when the population is finite and sampled items are not replaced. Use a binomial model when trials stay independent and the success probability remains constant.

4. What is the valid support range?

The support is the set of possible success counts. It runs from max(0, n − (N − K)) to min(n, K), preventing impossible event selections.

5. Why is the variance smaller than a similar binomial case?

Without replacement, draws become dependent. The finite population correction lowers dispersion because each observation reduces uncertainty about the remaining population.

6. What does the interval probability measure?

It adds probabilities across every success count from a to b. This is useful for statements like at least two successes or between one and three successes.

7. Can I use large population values?

Yes. The calculator uses logarithmic combination calculations, which helps keep large combinatorial terms numerically stable and practical for many finite sampling problems.

8. What do the CSV and PDF exports include?

The CSV export includes inputs, summary metrics, and the distribution table. The PDF export saves the displayed results section as a portable report snapshot.