Joint Probability Calculator — P(A ∩ B) for Independent & Dependent Events
Find the probability that two events occur together — P(A ∩ B) — for both independent and dependent events. Also computes the union P(A ∪ B), the conditional probabilities P(A|B) and P(B|A), and a breakdown of how the probability space is divided.
Event relationship
Probability that both A and B occur simultaneously
- 1
P(A)
40 % - 2
P(B)
30 % - 3
P(A ∩ B) = P(A) × P(B)
40 % × 30 % = 12 %For independent events the joint probability is the product of the two individual probabilities. - 4
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
40 % + 30 % − 12 % = 58 %
How does this calculator work?
For independent events, P(A∩B) = P(A) × P(B). For dependent events, P(A∩B) = P(A|B) × P(B). The union is P(A∪B) = P(A) + P(B) − P(A∩B). Enter both event probabilities (and P(A|B) for dependent events) to get all intersection, union, and conditional values.
Formula
How this is calculated
The joint probability P(A ∩ B) is the chance that two events A and B both occur. How you calculate it depends on whether the events are independent or dependent. Independent events do not influence each other — rolling a 6 on a die does not affect a separate coin flip. For independent events, P(A ∩ B) = P(A) × P(B). For example, if P(A) = 0.4 and P(B) = 0.3, then P(A ∩ B) = 0.12.
Dependent events are linked — the occurrence of one changes the probability of the other. In that case you need the conditional probability P(A|B), which is the probability of A given that B has already occurred. The multiplication rule gives P(A ∩ B) = P(A|B) × P(B). Bayes' theorem then lets you reverse the conditioning: P(B|A) = P(A ∩ B) / P(A).
The union P(A ∪ B) — the probability that at least one of A or B occurs — uses the inclusion-exclusion principle: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). The subtraction removes the double-counting of the joint region. The remaining regions are: only A (P(A) − P(A∩B)), only B (P(B) − P(A∩B)), and neither (1 − P(A∪B)).
Frequently asked questions
Joint probability P(A ∩ B) is the chance both events happen. Conditional probability P(A|B) is the chance A happens given that B has already happened. They are related by P(A ∩ B) = P(A|B) × P(B) — conditional probability factors out the knowledge that B occurred.
Two events are independent if knowing one occurred gives you no information about whether the other occurred — formally, P(A|B) = P(A) and P(B|A) = P(B). In practice, events from separate physical processes (rolling a die and flipping a coin) are independent; events from the same process (drawing cards without replacement) are dependent.
If A and B are mutually exclusive (they cannot happen at the same time), then P(A ∩ B) = 0 even though both have positive probability. For example, rolling a 3 and rolling a 5 on one die are mutually exclusive. Mutually exclusive events are a special case of dependent events — knowing one occurred means the other definitely did not.
TG we-Calculate Editorial Team. (2026). Joint Probability Calculator — P(A ∩ B) for Independent & Dependent Events [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/joint-probability-calculator
TG we-Calculate Editorial Team. "Joint Probability Calculator — P(A ∩ B) for Independent & Dependent Events." TG we-Calculate. 2026. https://we-calculate.com/calculator/joint-probability-calculator.
TG we-Calculate Editorial Team, "Joint Probability Calculator — P(A ∩ B) for Independent & Dependent Events," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/joint-probability-calculator
@misc{wecalculate_joint_probability_calculator, title = {Joint Probability Calculator — P(A ∩ B) for Independent & Dependent Events}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/joint-probability-calculator}}, year = {2026}, note = {TG we-Calculate} }
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