Probability, Statistics, and Data Analysis
Probability and statistics questions test whether you choose the right model, compute conditional probabilities correctly, and interpret data without overclaiming.
Core rules
- P(A∪B) = P(A) + P(B) − P(A∩B); mutually exclusive events have P(A∩B) = 0. Independence means P(A∩B) = P(A)P(B).
- Bayes: P(A|B) = P(B|A)P(A)/P(B), expanding P(B) over the partition of causes.
- Binomial: n fixed trials, constant p, P(k) = C(n,k)p^k(1−p)^(n−k). Poisson counts events at rate λ per interval, with mean = variance = λ. Exponential models time between Poisson events. Normal problems use z = (x−μ)/σ.
- Sample standard deviation divides by n−1; population divides by n. The central limit theorem makes x̄ approximately normal with standard error σ/√n.
- Confidence interval: x̄ ± z(α/2)σ/√n. Type I error is rejecting a true null (α); Type II is failing to reject a false null (β); power = 1 − β.
Worked example
Machine A makes 60% of parts with 2% defects; Machine B makes 40% with 5% defects. P(defect) = 0.6(0.02) + 0.4(0.05) = 0.032. Then P(B|defect) = 0.4(0.05)/0.032 = 0.625. So 62.5% of defects come from B, even though B produces fewer parts.
Traps
- Reporting P(defect|B) when the question asks for P(B|defect).
- Using σ instead of the standard error σ/√n.
- Applying the binomial to sampling without replacement from a small lot.
- Treating a p-value as the probability that the null hypothesis is true.