Probability
Core rules
- Sample space S lists all outcomes; event A is a subset. 0 ≤ P(A) ≤ 1 and P(S) = 1.
- Complement: P(A′) = 1 − P(A).
- Addition: P(A∪B) = P(A) + P(B) − P(A∩B); if mutually exclusive, the intersection term is zero.
- Conditional: P(A|B) = P(A∩B)/P(B), for P(B) > 0.
- Independent events: P(A∩B) = P(A)P(B), equivalently P(A|B) = P(A).
- For sampling without replacement, denominators fall after each draw.
Worked example
A bag has 5 red and 3 blue balls. Two are drawn without replacement. P(both red) = (5/8)×(4/7) = 20/56 = 5/14. The first draw leaves 7 balls and 4 red, so the second probability is 4/7, not 5/8. With replacement it would be (5/8)² = 25/64.
Traps
- Calling mutually exclusive events independent; they are different ideas.
- Using with-replacement denominators for without-replacement problems.
- For at least one, forgetting the complement shortcut 1 − P(none).
- Reversing P(A|B) and P(B|A).
- Adding probabilities of overlapping events without subtracting the intersection.