Probability and Probability Distributions
What this domain tests
Probability rules, conditional probability, independence and tree diagrams; the binomial and Poisson distributions; the normal distribution, including the normal approximation to the binomial and Poisson with continuity correction.
Core rules
- P(A∪B) = P(A) + P(B) - P(A∩B); P(A') = 1 - P(A).
- P(A|B) = P(A∩B)/P(B); independence means P(A∩B) = P(A)P(B).
- X ~ B(n, p): P(X = r) = C(n, r)p^r(1 - p)^{n-r}, mean np, variance np(1 - p).
- X ~ Po(λ): P(X = r) = e^{-λ}λ^r/r!, mean = variance = λ; rescale λ to match the interval used in the question.
- Normal: standardise z = (x - μ)/σ; sketch the curve to identify the correct tail before using the table.
Worked example
X ~ B(10, 0.3). Find P(X ≥ 2). Use the complement: P(X ≥ 2) = 1 - P(X = 0) - P(X = 1). P(0) = 0.7^10 ≈ 0.02825 and P(1) = 10(0.3)(0.7^9) ≈ 0.12106. So P(X ≥ 2) ≈ 1 - 0.14931 = 0.8507. For 'at least' questions the complement is almost always faster and safer.
Traps
- Assuming independence or mutual exclusivity without justification.
- Applying a Poisson λ to the wrong length of interval.
- Applying a continuity correction in the wrong direction: P(X ≥ 5) uses 4.5.
- Using the normal approximation before checking np and n(1 - p) are comfortably large.