TheCalculatory
Statistics · Probability Formula verified

Probability Calculator

Switch between three common probability questions: two independent events both happening, two mutually exclusive events happening, and counting combinations or permutations.

Question
%
050100
%
050100
On-screen number pad
Tap to type into the highlighted field
P(A and B)
25%
P(A) = 50%, P(B) = 50%
As a fraction
1 in 4
Complement
75%

Step-by-step proofCheck by hand

The three probability rules

Two independent events both happening multiplies their probabilities. Two mutually exclusive events — ones that can never both happen — happening either way adds their probabilities. And when a question is really about counting outcomes rather than combining given probabilities, combinations (order doesn't matter) and permutations (order does) count how many ways there are to choose r items from a set of n.

The formulas

P(A and B) = P(A) × P(B)  ·  P(A or B) = P(A) + P(B)
nCr = n! / (r!(n−r)!)  ·  nPr = n! / (n−r)!

Worked example

Two fair coin flips landing heads both times: P(A and B) = 0.5 × 0.5 = 25%. Choosing 2 items from a group of 6 without regard to order: 6C2 = 6! / (2!×4!) = 15 ways.

Frequently asked questions

What does 'independent' mean for two events?
Two events are independent when the outcome of one has no effect on the probability of the other — a coin flip landing heads doesn't change the odds of a separate die roll. For independent events, the probability of both happening is just the product of their individual probabilities.
What's the difference between a combination and a permutation?
Both count ways to choose r items from a group of n, but a permutation counts each different order separately, while a combination treats the same items in any order as one outcome. Choosing 2 letters from {A,B,C}: as combinations that's AB, AC, BC — 3 ways; as permutations, AB and BA count separately, giving 6 ways.
Why can't two mutually exclusive events' probabilities add up to more than 1?
Mutually exclusive means the two events can never both happen, so P(A or B) is just P(A) + P(B) with nothing to subtract for overlap. Since no probability can exceed 1, two mutually exclusive probabilities that sum past 1 describe an impossible situation — the inputs must be wrong.
Sources and method
  • Standard probability theory and combinatorics — general published mathematics.
Last reviewed: 20 Sep 2026 Sources last verified: 20 Sep 2026 Results use the assumptions explained on this page. Report an error How we check calculations

Related calculators

Standard Deviation Calculator Population and sample standard deviation, variance, and z-scores. 1 inputOpen → Mean, Median, Mode Calculator Mean, median, mode, midrange and quartiles from any list of numbers. 1 inputOpen → Quadratic Calculator Solve ax² + bx + c = 0 for its roots, with the vertex and step-by-step working. 3 inputsOpen →