Conditional probability

The formula

• P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0 — the probability of A given that B has happened.
• Equivalent product form: P(A ∩ B) = P(B)·P(A | B).

Worked examples

1. One die: P(odd | greater than 3). Greater than 3 = {4, 5, 6}, odd among them = {5} → 1/3.
2. An urn has 5 red and 3 blue balls. Two are drawn without replacement. P(second is red | first is red) = 4/7; P(both red) = 5/8 × 4/7 = 5/14.
3. In a class, 60% study maths, 40% study physics, 25% study both. P(physics | maths) = 0.25/0.60 = 5/12 ≈ 0.417.

Independence

• A and B are independent when P(A ∩ B) = P(A)·P(B), equivalently P(A | B) = P(A).
• In the example above, P(P) = 0.4 but P(P | M) ≈ 0.417, so the two subjects are not independent.
• Mutually exclusive and independent are different ideas: mutually exclusive events with non-zero probabilities are never independent.

Tree diagrams

• Multiply along a branch, add across branches.
• For two draws without replacement, the second-stage denominators are one less than the first.