Joint Probability Calculator
The probability that two events both happen — independent or not — with the conditional probabilities and an independence test.
For independent events P(A ∩ B) = P(A) P(B).
How the joint probability calculator works
For independent events P(A ∩ B) = P(A) P(B). When they are not independent you need a conditional: P(A ∩ B) = P(A | B) P(B). The calculator takes either route and reports both conditionals so you can see which direction the dependence runs.
The joint probability is always at most the smaller of the two marginals — you cannot have both more often than you have either.
Formula: P(A ∩ B) = P(A | B) P(B) = P(B | A) P(A)
Worked examples
| Inputs | P(A and B) | Note |
|---|---|---|
| Independent 40% and 25% | 10% | 10% |
| P(A|B) = 60%, P(B) = 25% | 15% | 15% |
| Two fair coins | 25% | 25% |
FAQFrequently asked questions
How do I find the probability of two things both happening?
Multiply, if they are independent. If not, multiply one by the other's conditional probability.
What does independent mean?
That knowing one happened does not change the odds of the other.
Can the joint exceed either single probability?
Never. "Both" is a subset of "either", so it is bounded by the smaller of the two.
What if the events are mutually exclusive?
Then the joint is zero — and they are as far from independent as it is possible to be.
Why does the order not matter?
Because P(A ∩ B) = P(B ∩ A). The two conditional routes give the same joint, which is exactly Bayes' theorem rearranged.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- National curriculum in England — Mathematics — the terms and methods taught in UK schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.