Multiplication rule and independence

Multiplication rule: the chance that two things both happen is

P(E ∩ F) = P(E) × P(F|E)

the chance of the first, times the chance of the second given the first.

This is exactly how drawing without replacement works. Two cards from a pack, both aces: P = 4/52 × 3/51 = 1/221. After one ace is gone, only 3 aces remain among 51 cards.

For three events: P(E ∩ F ∩ G) = P(E) × P(F|E) × P(G|E ∩ F). Keep updating the counts after each draw.

Independent events: knowing one tells you nothing about the other. Then P(F|E) = P(F), and the rule becomes

E and F are independent ⇔ P(E ∩ F) = P(E) × P(F)

Use this as the test. Example: one die. E = 'even' = {2, 4, 6}, F = {1, 2}. P(E) = 1/2, P(F) = 1/3, and E ∩ F = {2} has P = 1/6 = 1/2 × 1/3. So E and F are independent.

Do not confuse independent with mutually exclusive. Mutually exclusive means they cannot both happen: P(E ∩ F) = 0. If both events have positive chances, mutually exclusive events are dependent, since knowing one happened tells you the other did not.

Drawing with replacement, or tossing coins and dice again, gives independent trials.