Calculate joint, union, and conditional probabilities for two events, with contingency tables, Venn diagram visualization, lift analysis, and repeated-event chains.
The joint probability calculator computes P(A ∩ B) for two events, whether independent or dependent. It builds the full contingency table showing all possible combinations — both events occurring, only A, only B, or neither — and visualizes the overlap with a Venn diagram.
Joint probability is the foundation of multivariate statistics, Bayesian analysis, and machine learning. Understanding how two events combine — whether through independence (P(A∩B) = P(A)×P(B)) or dependence (P(A∩B) = P(A|B)×P(B)) — is critical for risk assessment, medical diagnosis, market basket analysis, and quality control.
This calculator also computes union probability, both conditional probabilities, lift (association strength), and the probability of repeated independent events in a chain. It is useful when you want to compare observed overlap against an independence baseline or see how quickly repeated trials change the chance of at least one match.
Joint probability connects simple AND/OR questions to dependence analysis, Bayesian reasoning, and multivariate models. This calculator makes overlap explicit with a contingency table, Venn diagram, conditional probabilities, and lift, so you can see how two events interact.
It is useful when comparing sampled data against an independence assumption, checking whether two outcomes co-occur more often than expected, or teaching the relationship between joint and conditional probability.
Independent: P(A ∩ B) = P(A) × P(B). Dependent: P(A ∩ B) = P(A|B) × P(B). Union: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Lift = P(A ∩ B) / [P(A) × P(B)].
Result: P(A ∩ B) = 12%, P(A ∪ B) = 58%
For independent events with P(A) = 0.3 and P(B) = 0.4, the joint probability is 0.3 × 0.4 = 0.12 (12%), and the union is 0.3 + 0.4 − 0.12 = 0.58 (58%).
Joint distributions over multiple variables form the backbone of Bayesian networks. A Bayesian network factors the joint distribution P(A,B,C,...) into conditional probabilities along a directed acyclic graph. Understanding two-event joint probability is the first step toward these powerful models.
Market basket analysis uses joint probability concepts extensively. Support = P(A ∩ B), confidence = P(A|B), and lift = P(A ∩ B) / [P(A)×P(B)]. Rules with high support, confidence, and lift identify meaningful product associations in retail data.
The chi-square test for independence compares observed joint frequencies to expected frequencies under independence (P(A)×P(B)). Large deviations signal dependence. This calculator's lift metric provides the same insight for two events — lift ≠ 1 suggests dependence.
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Joint probability P(A ∩ B) is the probability both A and B occur. Conditional probability P(A|B) is the probability of A assuming B has already occurred. They're related: P(A ∩ B) = P(A|B) × P(B).
Events are independent if knowing one occurred doesn't change the probability of the other: P(A|B) = P(A). Equivalently, P(A ∩ B) = P(A) × P(B). This calculator tests this with the lift metric.
Lift measures the strength of association between two events. It's the ratio of actual joint probability to what it would be under independence. Lift of 2 means the events co-occur twice as often as expected by chance.
No. Probabilities are always between 0 and 1. Joint probability P(A ∩ B) is the "overlap" in the Venn diagram and is always ≥ 0.
For binary events, correlation is directly related to lift and joint probability. Positive correlation (lift > 1) means events tend to occur together. Phi coefficient provides a normalized measure for 2×2 tables.
That's impossible. The calculator constrains inputs to ensure valid probability values. If you enter P(A|B) > 1 effectively, the calculator will cap the result.