Conditional probability describes how the probability of an event updates when additional evidence is observed, and correct reasoning requires conditioning on *all* available evidence rather than a partial subset of it — a subtlety illustrated by the Monty Hall problem, where the host's deliberate action of revealing a goat carries information beyond the mere fact that a goat was revealed. Simpson's paradox demonstrates that an inequality holding within every stratum of a confounding variable can reverse when the strata are aggregated, because the relative weights (conditional probabilities of the confounder) differ across the groups being compared. Both phenomena sit within probability theory's broader treatment of conditioning, the law of total probability, and the distinction between conditional and unconditional (marginal) probabilities.
Conditional probability describes how the probability of an event updates when additional evidence is observed, and correct reasoning requires conditioning on *all* available evidence rather than a p…