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General Multiplication Rule

P(A∩B)=P(A) P(B∣A)P(A \cap B) = P(A) \, P(B \mid A)

Worked example: Car 0.6, car-and-bike 0.15 → P(bike|car) = 0.25 — press Try an example to run it live, then adjust anything.

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General Multiplication Rule explained

P(A)P(B|A)P(A∩B)

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.

General Multiplication Rule formula

P(A∩B)=P(A) P(B∣A)P(A \cap B) = P(A) \, P(B \mid A)
Where
  • P(A∩B)P(A \cap B)= Probability both occur
  • P(A)P(A)= Probability of A
  • P(B∣A)P(B \mid A)= Probability of B given A

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