Union, Intersection & Complement Calculator
Find the union, intersection, and complement probabilities for two events, treating them as independent or using a supplied joint probability.
Relationship
Probability that A or B (or both) occur
- 1
P(A∩B) = P(A) × P(B)
0.5 × 0.4 = 0.2Events are independent, so the joint probability is the product. - 2
P(A∪B) = P(A) + P(B) − P(A∩B)
0.5 + 0.4 − 0.2 = 0.7000
How does this calculator work?
For two events, the complement is 1 − P. The union is P(A) + P(B) − P(A∩B), subtracting the overlap once. The intersection is P(A)·P(B) for independent events, or a supplied joint value otherwise. Enter each probability in [0, 1]; the joint value must not exceed the smaller marginal.
Formula
How this is calculated
Enter the marginal probabilities P(A) and P(B), each a value between 0 and 1. The relationship dropdown controls how the intersection P(A∩B) is found: choose "Independent" to compute it as the product P(A∩B) = P(A)·P(B), or choose "Enter P(A∩B) directly" to type a known joint probability for events that may be dependent.
The complement rule gives P(Aᶜ) = 1 − P(A) and P(Bᶜ) = 1 − P(B), the probabilities that each event does not occur. The addition rule gives the union P(A∪B) = P(A) + P(B) − P(A∩B); the intersection is subtracted once because it is otherwise counted in both P(A) and P(B). All outputs are dimensionless probabilities in [0, 1].
Validation enforces that every probability lies in [0, 1], that a supplied joint probability never exceeds min(P(A), P(B)) (an intersection cannot be larger than either event), and that the resulting union does not exceed 1. If any constraint is violated the calculator reports invalid input rather than an impossible result.
Frequently asked questions
Only when the events are independent, meaning the occurrence of one does not change the probability of the other. For dependent events you must supply the joint probability P(A∩B) directly.
Adding P(A) and P(B) counts the overlap (both events occurring) twice, so the intersection is subtracted once to count it a single time.
The intersection cannot exceed either marginal probability, so P(A∩B) must be ≤ min(P(A), P(B)), and the union cannot exceed 1. Values breaking these rules describe impossible event combinations.
Also known as
TG we-Calculate Editorial Team. (2026). Union, Intersection & Complement Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/event-probability-calculator
TG we-Calculate Editorial Team. "Union, Intersection & Complement Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/event-probability-calculator.
TG we-Calculate Editorial Team, "Union, Intersection & Complement Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/event-probability-calculator
@misc{wecalculate_event_probability_calculator, title = {Union, Intersection & Complement Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/event-probability-calculator}}, year = {2026}, note = {TG we-Calculate} }
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