Union & Intersection Probability Calculator

How it is calculated

Enter P(A), P(B) and the overlap P(A ∩ B) as percentages. The tool applies the addition rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B) and also returns the non-overlapping parts only A = P(A) − P(A ∩ B) and only B = P(B) − P(A ∩ B). Results are percentages with 2 decimal places. The overlap must be consistent with the other two numbers. It cannot be negative or larger than the smaller of P(A) and P(B), and by inclusion-exclusion it cannot be below P(A) + P(B) − 100, otherwise the union would exceed 100 %. Inconsistent inputs, values outside 0–100 and empty boxes give no result. Unlike the independent-events tool, no independence is assumed here, because you supply the overlap yourself.

Common mistakes

• Adding P(A) and P(B) and stopping. 40 % + 30 % = 70 % counts the 10 % overlap twice, so the true union is 60 %.
• Entering the probability of both as the probability of only one of them. The third box is the overlap itself, and "only A" is calculated for you.
• Entering an overlap larger than either event, for example 35 % when P(B) is 30 %. The overlap cannot exceed the smaller event, and the tool returns nothing.
• Entering an overlap that is too small. With P(A) = 80 % and P(B) = 80 % the overlap must be at least 60 %; a value of 10 % is impossible and is refused.
• Using the product P(A) × P(B) as the overlap without checking independence. If the events are dependent, the product is wrong and the union will be off.