Probability: Random Experiments
This domain fixes the vocabulary and the addition rule on which the rest of probability builds. Expect questions on sample spaces, equally likely outcomes, mutually exclusive and exhaustive events, independence, and set operations on events.
Core rules
- A random experiment has a known set of outcomes but an unpredictable result; the sample space S lists all outcomes.
- For equally likely outcomes, P(A) = number favourable to A ÷ total outcomes in S.
- Axioms: 0 ≤ P(A) ≤ 1; P(S) = 1; for mutually exclusive events P(A ∪ B) = P(A) + P(B).
- General addition theorem: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
- Complement: P(A′) = 1 − P(A); P(A ∩ B′) = P(A) − P(A ∩ B).
- Independence: P(A ∩ B) = P(A)P(B), equivalently P(A|B) = P(A).
- De Morgan: (A ∪ B)′ = A′ ∩ B′, and (A ∩ B)′ = A′ ∪ B′.
Worked example
Let P(A) = 0.4 and P(B) = 0.5, with A and B independent. Then P(A ∩ B) = 0.2, so P(A ∪ B) = 0.4 + 0.5 − 0.2 = 0.7. The probability of exactly one occurring is P(A)P(B′) + P(A′)P(B) = (0.4)(0.5) + (0.6)(0.5) = 0.5.
Traps that cost marks
Adding probabilities without subtracting the intersection; treating mutually exclusive events as independent (they cannot be, unless one has probability zero); assuming outcomes are equally likely when the sample space has been listed carelessly; forgetting that “at least one” is usually quickest via the complement; ignoring whether selection is with or without replacement.