Joint Probability Calculator
Combine marginal probabilities and conditional information to find P(A ∩ B). The tool also reports unions, complements, and derived conditional probabilities.
Leave blank if unknown.
Leave blank if unknown.
Joint Probability
15.00%
P(A ∩ B)
Union
55.00%
P(A ∪ B)
Neither Event
45.00%
Probability neither occurs
| Outcome | Probability |
|---|---|
| Only A | 25.00% |
| Only B | 15.00% |
| Both A and B | 15.00% |
| Neither | 45.00% |
Derived conditionals:
P(A | B) = 50.00%
P(B | A) = 37.50%
How to Use This Calculator
- Enter the marginal probabilities P(A) and P(B).
- Optionally enter conditional probabilities P(A | B) and/or P(B | A).
- Tick the independence box if events are independent and no conditional information is available.
- Review the joint probability, unions, and derived conditionals. Warnings indicate inconsistent assumptions.
Formula
P(A ∩ B) = P(A | B) · P(B) = P(B | A) · P(A)
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
If independent: P(A ∩ B) = P(A) · P(B)
The calculator reconciles any provided relationships and flags contradictions to guide you toward consistent probability models.
Full Description
Joint probabilities evaluate the likelihood of two events occurring together. They are central to risk analysis, contingency tables, and Bayesian reasoning. With marginal and conditional probabilities, you can derive the full joint distribution and understand overlaps and mutual exclusivity.
This tool helps students and practitioners cross-check assumptions, detect inconsistent inputs, and translate between perspectives (e.g., from P(A | B) to P(B | A)) using Bayes' theorem algebra.
Frequently Asked Questions
What if I only know P(A) and P(B)?
Without additional information, the joint probability is undefined; you must assume independence or supply conditional probabilities.
Why do I see warnings about inconsistency?
If P(A | B) · P(B) and P(B | A) · P(A) differ significantly, your inputs conflict. Adjust the values or re-express them to match.
Can probabilities exceed 1?
No. The calculator clamps results to [0, 1] and warns if intermediate calculations stray outside that range, highlighting issues with the inputs.
How is this different from the conditional probability calculator?
This page focuses on deducing joint probabilities and unions, while the conditional probability calculator starts from a Venn-diagram perspective and emphasizes complements. Use whichever framing suits your problem.