Basic probability

Definitions

• The sample space S is the set of all possible outcomes; an event A is a subset of S.
• P(A) = number of favourable outcomes / number of equally likely outcomes.
• 0 ≤ P(A) ≤ 1; P(S) = 1 and P(∅) = 0.
• Complement rule: P(A′) = 1 − P(A).

Addition rule

• P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
• If A and B are mutually exclusive, P(A ∩ B) = 0 and P(A ∪ B) = P(A) + P(B).
• Example: one die — P(even or greater than 4) = 3/6 + 2/6 − 1/6 = 4/6 = 2/3.

Multiplication rule

• For independent events, P(A ∩ B) = P(A)·P(B).
• Example: two fair coins — P(two heads) = ½ × ½ = ¼.
• Without replacement the second probability changes: two cards drawn from a pack, P(both aces) = 4/52 × 3/51 = 1/221.