What Is Combination Probability?
Combination probability counts equally likely selections. If N items contain K special ones and you choose r of them, the probability of exactly k special items is C(K, k) × C(N − K, r − k) ÷ C(N, r).
Enter N, K, r and k to get the exact probability, the cumulative at most and at least values, the expected number of special items and a bar chart across all possible k.
This is the hypergeometric setting: order does not matter and nothing is put back. Use the binomial instead if each pick is independent.
How to Use the Combination Probability Calculator
- Enter N, K and r.
- Enter the number of special items you want.
- Read the exact, at most and at least probabilities.
- Check the combination counts and chart.
Combination Probability Formula
Combinations
Probability
Combination Probability Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| N | Total items | |
| K | Special items among them | |
| r | Items chosen | |
| k | Special items wanted in the choice | |
| C(n, r) | Combinations, n choose r |
Combination Probability Worked Example
A 5-card hand from a 52-card deck has exactly 2 hearts with probability C(13, 2) × C(39, 3) ÷ C(52, 5) = 78 × 9139 ÷ 2598960 = 0.2743.
With the example values loaded in the calculator above, the steps are:
- Ways to pick 2 of the 13 special items: C(13, 2).
- Ways to pick the other 3 from the 39 ordinary items: C(39, 3).
- All ways to pick 5 items from 52: C(52, 5).
- P = C(13, 2) × C(39, 3) ÷ C(52, 5) = 0.27427971.
Result: P(exactly 2 special) = 0.27427971; P(at least 2 special) = 0.36704682.
How to Read a Combination Probability
The result is a share of equally likely selections. A 5-card hand from a 52-card deck can be chosen in C(52, 5) = 2,598,960 ways. Of those, 78 × 9,139 = 712,842 contain exactly 2 hearts, so the probability is 712,842 ÷ 2,598,960 ≈ 0.2743.
The expected number of special items is r × K ÷ N, which is 5 × 13 ÷ 52 = 1.25 hearts. The full distribution table shows how the probability spreads across 0, 1, 2, 3, 4 and 5 hearts, so you can compare "exactly" with "at most" and "at least".
Where Combination Probability Is Used
Card hands
Find the chance of a hand with a given number of hearts, aces or face cards.
Quality control
Estimate the chance that a sample from a batch contains exactly k defective items.
Lotteries and draws
Work out the chance of matching k winning numbers when numbers are drawn without replacement.
Selecting groups
Find the chance that a randomly chosen committee has a set number of members from one group.
Common Combination Probability Mistakes to Avoid
- The model assumes selection without replacement. If items are put back, use the binomial distribution.
- Order does not matter here. If order matters, count permutations instead.
- Check that k cannot exceed K or r, and that r − k cannot exceed N − K. Impossible cases have probability 0.
- Do not confuse N, the total, with n, the sample. The calculator uses N, K, r and k as labelled.
Combination Probability Terms and Definitions
- Combination
- A selection where order does not matter.
Combination Probability FAQs
How do you find probability using combinations?
Count the favourable selections and divide by all possible selections. For exactly k special items it is C(K, k) × C(N − K, r − k) ÷ C(N, r).
What if order matters?
Then count ordered arrangements with permutations, nPr, in both the numerator and the denominator. For this kind of "how many of each type" question the order usually does not matter.
How is this different from the binomial?
The binomial assumes independent trials with a fixed probability, as with replacement. This hypergeometric setting samples without replacement, so each draw changes the chance of the next. The two agree closely when N is much larger than r.
What is the expected number of special items?
It is r × K ÷ N. Drawing 5 cards from a deck with 13 hearts gives 5 × 13 ÷ 52 = 1.25 hearts on average.