Probability
Permutations & combinations
About This Calculator
Probability measures the likelihood of an event occurring, expressed as a value between 0 (impossible) and 1 (certain). Permutations count ordered arrangements of items, while combinations count unordered selections. These are the building blocks of probability theory, statistics, and combinatorics.
Formula
Probability: P(event) = favorable outcomes / total outcomes
Permutation (ordered): P(n,r) = n! / (n−r)!
Combination (unordered): C(n,r) = n! / (r! × (n−r)!)
Complement: P(not A) = 1 − P(A)
Example Calculation
A bag has 5 red and 3 blue balls. What is the probability of drawing a red ball?
- Total outcomes = 5 + 3 = 8
- Favorable outcomes (red) = 5
- P(red) = 5/8 = 0.625 = 62.5%
P(red) = 62.5%
Probability of Common Events
| Event | Probability | Fraction |
|---|---|---|
| Coin heads | 50% | 1/2 |
| Die rolls a 6 | 16.7% | 1/6 |
| Two heads in a row | 25% | 1/4 |
| At least one head (2 flips) | 75% | 3/4 |
| Royal flush (poker) | 0.000154% | 1/649,740 |
Frequently Asked Questions
What is the difference between permutation and combination?
A permutation is an ordered arrangement — the order matters. A combination is an unordered selection — the order doesn't matter. For example, choosing 2 from {A,B,C}: permutations are AB,BA,AC,CA,BC,CB (6); combinations are AB,AC,BC (3).
What does mutually exclusive mean?
Two events are mutually exclusive if they cannot both happen at the same time. For example, rolling a 2 and rolling a 5 on one die. For mutually exclusive events, P(A or B) = P(A) + P(B).
What is conditional probability?
P(A|B) is the probability of A given that B has already occurred. It equals P(A and B) / P(B). For example, drawing two aces from a deck: P(2nd ace | 1st ace drawn) = 3/51.
What does independent event mean?
Two events are independent if the outcome of one does not affect the other. For independent events, P(A and B) = P(A) × P(B). Coin flips are independent; drawing cards without replacement is not.