Probability Exactly One Event Calculator

Enter P(A), P(B), and their intersection P(A and B) as decimal probabilities from zero to one. The calculator removes the shared outcomes from both event probabilities, returns each exclusive part, and adds them to produce the final result. It works for dependent, independent, overlapping, or mutually exclusive events because it uses the intersection you supply instead of assuming a relationship.

Understand what exactly one event means

Exactly one event means A occurs and B does not, or B occurs and A does not. It excludes outcomes where neither event occurs and outcomes where both occur together. On a Venn diagram, the desired region consists of the two nonoverlapping crescents, not the shared center. Start with P(A), but remove P(A and B) to obtain P(A only). Do the same with P(B) to obtain P(B only). Because those two exclusive regions cannot overlap, their probabilities can then be added. Combining the steps gives P(exactly one) = P(A) + P(B) − 2 × P(A and B). The intersection is subtracted twice because it appeared once inside each marginal probability, while it must appear zero times in the answer. This differs from the probability of at least one event, where the intersection is subtracted only once and remains included. It also differs from an exclusive-or statement written casually in everyday language, where the intended treatment of both events may be unclear. This calculator uses the strict probability definition: one event occurs, and only one.

Enter valid probabilities and interpret the output

Provide all three inputs as finite decimal numbers in the closed interval from zero to one. Write sixty percent as 0.6 and five percent as 0.05; do not enter a percent sign or a whole-number percentage. The intersection must describe both events occurring in the same experiment, population, and time period used for the two marginal probabilities. A valid intersection cannot exceed P(A) or P(B), because shared outcomes are contained within each event. It also cannot be so small that P(A or B) would exceed one. The calculator checks these probability rules rather than returning a plausible-looking number from an impossible combination. The response repeats the accepted inputs and reports P(A only), P(B only), and P(exactly one), followed by the formula. For example, if P(A) is 0.6, P(B) is 0.5, and P(A and B) is 0.2, then P(A only) is 0.4 and P(B only) is 0.3. Adding the exclusive parts gives 0.7. Boundary values are valid: an answer of zero means there are no outcomes belonging to only one event, while an answer of one means exactly one of the events always occurs.

Use the supplied intersection without hidden assumptions

The calculation does not require A and B to be independent. Independence is a specific relationship that would let you derive the intersection as P(A) × P(B), but this capability deliberately accepts P(A and B) directly. Use the intersection stated by your table, experiment, survey, model, or exercise. Likewise, do not set the intersection to zero unless the events truly are mutually exclusive. Doing so when overlap exists inflates the exactly-one probability because shared outcomes are incorrectly assigned to both exclusive regions. Before automating a calculation, confirm that all three values use the same denominator and event definitions. A monthly risk and an annual risk, or proportions taken from different populations, do not form a meaningful set merely because each number is individually between zero and one. The result can support survey segmentation, reliability studies, medical test summaries, quality-control rules, and probability coursework, but it describes only the events you define. It does not infer causation or dependence. The deterministic calculation uses no outside data, network calls, randomness, or estimation. Browser use is free, while an automated API request costs $0.002; repeated identical inputs therefore always produce the same output.