Conditional probability formalizes the likelihood of an event given that another event has occurred, defined as the probability of both events jointly divided by the probability of the conditioning event, which restricts the sample space to the outcomes consistent with the conditioning event. From this definition follow the product rule (and its generalization to multiple events) for computing joint probabilities, and the notion of "a posteriori" conditional probabilities, where the conditioning and target events are reversed in temporal order but computed with the same formula. This material belongs to conditional probability theory within probability and statistics, and it directly explains counterintuitive phenomena such as the base rate fallacy, where a diagnostic test's positive-predictive value can be far lower than its apparent accuracy suggests when the base rate of the underlying condition is low, and Simpson's paradox, where aggregate and disaggregated conditional comparisons can reverse direction.
Conditional probability formalizes the likelihood of an event given that another event has occurred, defined as the probability of both events jointly divided by the probability of the conditioning e…