Multiplication theorem on probability

E ∩ F, also written EF, is the event that both E and F occur.

Clearing the fraction in the definition of conditional probability gives the multiplication rule: P(E ∩ F) = P(E) P(F|E), and equally P(F) P(E|F). The first form needs P(E) ≠ 0, the second P(F) ≠ 0.

Read it as a sequence: the chance that E happens, times the chance that F then happens once E has.

An urn holds 10 black and 5 white balls; two are drawn in turn without replacement. P(first black) = 10/15. With one black gone, P(second black | first black) = 9/14. So P(both black) = (10/15)(9/14) = 3/7.

For three events: P(A ∩ B ∩ C) = P(A) P(B|A) P(C|A ∩ B). The rule extends the same way to four or more events.

Three cards are drawn in turn without replacement from a well-shuffled pack of 52. P(king, king, ace in that order) = (4/52)(3/51)(4/50) = 2/5525.

Without replacement, each draw changes the counts, so every factor after the first is a conditional probability. With replacement the counts are restored and the factors do not change.