Conditional Probability Calculator P(A|B)

Conditional probability is one of the cornerstones of probability theory and statistics. It describes the likelihood of an event occurring given that another event has already happened, and it underpins some of the most important reasoning tools in science, medicine, and machine learning. The formal definition is: P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0. Here, P(A|B) is read as 'the probability of A given B', P(A ∩ B) is the joint probability that both A and B occur, and P(B) is the marginal probability of B. Rearranging this formula gives the multiplication rule: P(A ∩ B) = P(A|B) × P(B), which is widely used to compute joint probabilities from conditional ones. A classic illustration is medical testing. Suppose a disease affects 1% of the population and a diagnostic test has a 5% false-positive rate. The probability that a randomly selected person tests positive is P(B). The probability that the person is both diseased and tests positive is P(A ∩ B). Dividing gives the conditional probability that a person is actually sick given a positive result — often far lower than intuition suggests, a phenomenon known as the base rate fallacy. Conditional probability also lies at the heart of Bayes' theorem: P(A|B) = P(B|A) × P(A) / P(B). Bayes' theorem lets you update your prior belief P(A) in light of new evidence B to obtain a posterior belief P(A|B). This Bayesian updating framework is used in spam filters, medical diagnosis, forensic evidence evaluation, and modern machine learning classifiers. This calculator supports three modes. 'Find P(A|B)' takes the joint probability P(A ∩ B) and the marginal P(B) as inputs and returns the conditional probability. 'Find P(A ∩ B)' takes P(A|B) and P(B) and applies the multiplication rule. 'Find P(B)' solves for the marginal probability given the conditional and joint values. All probability inputs must be between 0 and 1, and P(B) must be non-zero when it appears in the denominator.