Probability and Probability Distributions
Probability questions test the rules of combination, conditional probability and the standard distributions. You must decide whether events are independent, whether sampling is with or without replacement, and whether a discrete or continuous model fits.
Core rules
- Addition: P(A or B) = P(A) + P(B) − P(A and B). For mutually exclusive events the last term is zero.
- Multiplication: P(A and B) = P(A)P(B|A); independent events give P(A)P(B).
- Conditional probability: P(A|B) = P(A and B)/P(B). Bayes theorem reverses the conditioning.
- Binomial: n fixed trials, two outcomes, constant p, independent trials; P(X = r) = C(n,r)p^r(1−p)^(n−r), mean np, variance np(1−p).
- Poisson: counts in an interval with mean λ; P(X = r) = e^−λ λ^r/r!. Scale λ if the interval changes.
- Normal: standardise z = (x − μ)/σ; use the complement for upper tails.
Worked example
A process has a 10% defective rate. In a sample of 3 items, find the probability of at least one defective. At least one is the complement of none. P(none) = 0.9³ = 0.729. Therefore P(at least one) = 1 − 0.729 = 0.271. A direct calculation would add P(1) + P(2) + P(3); the complement is faster and less error-prone. The binomial model is valid only if items are independent and the defect rate is constant.
Traps
- Adding probabilities of non-mutually-exclusive events without subtracting the overlap.
- Using the binomial when sampling without replacement from a small population.
- Forgetting to scale the Poisson mean for a different time or area.
- Treating P(A|B) as P(B|A).