General Multiplication Rule

This is the honest version of "multiply the probabilities": the second factor is the chance of B after A has already happened. Independence is only the special case where P(B|A) = P(B). Drawing two aces from a shuffled deck without replacement, P = (4/52) × (3/51) = 12/2652 ≈ 0.00452, one chance in 221, because the first ace leaves 51 cards holding 3 aces. Enter probabilities as decimals from 0 to 1, or switch the unit to %.

Edward Thorp built Beat the Dealer (1962) on precisely this dependence: in blackjack the cards already dealt change the odds on the next one, so the deck acquires a memory that roulette wheels lack, and casinos answered by shuffling several decks together. The trap is reusing the unconditional figure for the second draw: 4/52 twice gives 0.0059, about 30% too high. Rearranged, the rule extracts the conditional from survey data: if 60% of customers own a car and 15% own both a car and a bike, then P(bike | car) = 0.15/0.6 = 0.25.