Probability and Probability Distributions
What the exam tests
Probability rules, conditional probability and Bayes' theorem, random variables, and standard distributions - binomial, Poisson and normal. Expect event algebra and computation of probabilities.
Core rules
- Sample space S; 0 ≤ P(A) ≤ 1; P(S) = 1. Complement: P(A′) = 1 − P(A).
- Addition: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Mutually exclusive events have P(A ∩ B) = 0.
- Multiplication: P(A ∩ B) = P(A)P(B|A); independent events have P(A ∩ B) = P(A)P(B).
- Conditional: P(A|B) = P(A ∩ B)/P(B), P(B) > 0. Bayes: P(Ai|B) = P(Ai)P(B|Ai)/ΣP(Aj)P(B|Aj).
- Binomial B(n, p): P(X = r) = nCr p^r q^(n−r); mean np; variance npq.
- Poisson: P(X = r) = e^(−λ)λ^r/r!; mean = variance = λ.
- Normal N(μ, σ²): symmetric about μ; z = (x − μ)/σ; use the standard normal table.
Worked example
A fair die is rolled twice. P(sum = 7) = 6/36 = 1/6, from the six favourable pairs. For at least one six, use the complement: 1 − (5/6)² = 11/36. Binomial check: n = 5, p = 0.2, so P(X = 2) = 5C2(0.2)²(0.8)³ = 10 × 0.04 × 0.512 = 0.2048.
Traps
Adding probabilities of non-mutually-exclusive events without subtracting the overlap; treating independent events as mutually exclusive; computing at least one directly instead of using the complement; forgetting that Poisson approximates binomial when n is large and p is small; confusing σ with σ² in z-scores; using a one-tail critical value for a two-tail problem.