Probability Combination Calculator

Enter Calculator Inputs

Calculation Mode

Population Size (N)

Success States (K)

Draws (n)

Target Successes (x)

Probability Range

Results appear below the header and above this form after submission.

Example Data Table

Scenario	Population	Success States	Draws	Target Successes	Range	Probability
Cards drawing aces	52	4	5	2	Exactly	0.039930
Defective items sample	100	8	10	1	At least	0.583752
Committee selection	12	5	4	2	At most	0.858586

Formula Used

The calculator uses combinations without order. It also uses the hypergeometric probability model for draws without replacement.

Combination:
C(n, r) = n! / (r! (n - r)!)

Exact probability without replacement:
P(X = x) = [C(K, x) × C(N - K, n - x)] / C(N, n)

Where:
N = total population
K = total success states
n = total draws
x = target successes

Direct probability:
P(Event) = Favorable combinations / Total combinations

At least and at most probabilities are calculated by summing valid exact probabilities across the required success range.

How to Use This Calculator

Select the calculation mode that matches your problem.
Enter population, success states, draws, and target successes for sampling problems.
Choose exactly, at least, or at most for the probability range.
Use direct mode when you already know favorable and total combinations.
Press submit to display results above the form.
Download the result as CSV or PDF for records.

Frequently Asked Questions

1. What does this calculator solve?

It calculates combinations and probabilities for selection problems. It supports direct event probability and sampling without replacement using combination-based formulas.

2. What is a combination?

A combination counts selections where order does not matter. Choosing A, B, C is the same as choosing C, B, A.

3. When should I use without replacement mode?

Use it when each item can be drawn only once. Card draws, sampled products, and committee picks usually follow this rule.

4. What is the difference between exactly, at least, and at most?

Exactly means one precise success count. At least sums that count and above. At most sums that count and below.

5. Can I use this for cards and lottery problems?

Yes. It works well for cards, balls in urns, defect sampling, committee building, and other discrete selection problems.

6. Why do results show many decimals?

Probability values are often small. Extra decimals help compare events accurately, especially when two outcomes look close.

7. What happens if my inputs are impossible?

The calculator checks input limits first. It shows an error when draws exceed population, favorable cases exceed totals, or target values are invalid.