Probability and probability distributions
Probability and probability distributions
This domain tests probability rules, conditional probability and Bayes' theorem, random variables, and the binomial, Poisson and normal distributions. The emphasis is on choosing the correct model and interpreting parameters.
Core rules
- Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Conditional probability: P(A|B) = P(A ∩ B)/P(B), provided P(B) > 0.
- Events A and B are independent if P(A ∩ B) = P(A)P(B), equivalently P(A|B) = P(A).
- For a discrete random variable, E(X) = ΣxP(X = x) and Var(X) = E(X²) − [E(X)]².
- Binomial B(n, p): fixed n independent trials, constant p, P(X = r) = C(n,r)p^r(1−p)^(n−r); mean np, variance np(1−p).
- Poisson: independent events in a fixed interval with constant mean rate λ; P(X = r) = e^(−λ)λ^r/r!; mean = variance = λ.
- Normal: standardise with z = (x − μ)/σ; use continuity correction when approximating a discrete distribution.
Worked example
If X ~ B(10, 0.2), then P(X ≥ 1) = 1 − P(X = 0) = 1 − 0.8¹⁰ = 1 − 0.1074 = 0.8926.
Common traps
Using the binomial when sampling is without replacement; omitting the conditional denominator; using σ² instead of σ in normal standardisation; forgetting continuity correction; and failing to rescale a Poisson rate for a different interval.