Probability and Statistics
Core ideas
Distinguish discrete and continuous random variables. For discrete variables, use probability distributions summing to one, and compute E(X) = ΣxP(x) and Var(X) = E(X²) − [E(X)]². Know the binomial distribution: n independent trials, constant p, X ~ Bin(n, p), with P(X = k) = C(n,k)pᵏ(1−p)^{n−k}. For continuous variables, use the normal distribution with standardisation z = (x − μ)/σ and tables or technology. Interpret standard deviation as spread, and describe data with mean, median, quartiles and outliers. Understand conditional probability P(A|B) = P(A∩B)/P(B) and independence.
Traps
- Using binomial when trials are not independent or p changes.
- Forgetting the continuity correction is not needed for exact binomial, but normal approximations require care.
- Computing Var(X) as E(X²) − E(X) without squaring the mean correctly.
- Confusing P(A|B) with P(B|A).
- Ignoring the upper tail when finding normal probabilities.
Worked example
A fair die is rolled 10 times. Find P(exactly 3 sixes). X ~ Bin(10, 1/6). P(X = 3) = C(10,3)(1/6)³(5/6)⁷ ≈ 0.155. Rule of thumb: write the distribution and parameters before computing, and check that probabilities lie between 0 and 1.