How to use the Probability Calculator
- Enter the probability of each event as a decimal or percentage.
- State whether the events are independent or mutually exclusive.
- Read P(A and B), P(A or B), and the complements.
- Use the conditional field when one event's occurrence changes the other's odds.
How the calculation works
For independent events P(A and B) = P(A) × P(B). For any two events P(A or B) = P(A) + P(B) − P(A and B), where the subtraction avoids double-counting the overlap; with mutually exclusive events that overlap is zero. The complement rule, P(not A) = 1 − P(A), is often the fastest route: 'at least one' problems are almost always easier as 1 minus the probability of none.
Conditional probability is P(A|B) = P(A and B) ÷ P(B), and confusing it with P(B|A) is the classic error behind misread medical tests. With a rare condition, a highly accurate test still produces mostly false positives, because the false positives from the large healthy population outnumber the true positives from the small affected one. Bayes' theorem formalises the correction.
P(A∩B) = P(A)P(B) if independent ; P(A∪B) = P(A) + P(B) − P(A∩B) ; P(A|B) = P(A∩B)/P(B)Source: Kolmogorov probability axioms with standard addition, multiplication and conditional probability rules.
Worked example
A disease affects 0.5% of people. A test is 98% sensitive with a 3% false-positive rate. You test positive — what is the chance you have it?
- True positives = 0.005 × 0.98 = 0.0049.
- False positives = 0.995 × 0.03 = 0.02985.
- Total positives = 0.03475.
- P(disease | positive) = 0.0049 / 0.03475 = 14.1%.
Only about 14% — the base rate dominates, which is why confirmatory testing exists.
Frequently asked questions
What is the difference between independent and mutually exclusive?+
Independent means one does not affect the other's probability. Mutually exclusive means they cannot both happen.
Why is 'at least one' easier via the complement?+
Because 1 − P(none) needs one calculation, while enumerating every 'at least one' case needs many.
Can probabilities be added directly?+
Only for mutually exclusive events. Otherwise you double-count the overlap.
What is the base rate fallacy?+
Ignoring how rare a condition is when interpreting a positive test, which vastly overstates the chance of having it.
Last reviewed August 31, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.