Three classic paradoxes — seemingly mad, in reality perfectly consistent with the conditional probability we have just seen. They serve to build the right instinct: in probability the natural answer is almost always wrong, and Bayes’ formula is the only reliable arbiter. The Monty Hall paradox shows how the host’s information shifts the probabilities; the two-children one how the precise wording changes the answer; the Bertrand one how “at random” is an empty notion without a distribution.