Probability & Statistics
What Probability & Statistics tests
The Probability & Statistics papers test data summary and presentation, permutations and combinations, probability rules, discrete random variables, binomial and normal distributions, correlation and regression, and hypothesis testing. Questions reward choosing the right model and interpreting parameters in context, not just substituting into a formula.
Core rules
- For grouped data, histogram frequency density = frequency ÷ class width; use midpoints for estimates of the mean.
- Permutations count ordered arrangements, nPr = n!/(n−r)!; combinations count selections, nCr = n!/[r!(n−r)!].
- P(A∪B) = P(A) + P(B) − P(A∩B); for independent events P(A∩B) = P(A)P(B); P(A|B) = P(A∩B)/P(B).
- For a discrete random variable, E(X) = Σxp and Var(X) = E(X²) − [E(X)]².
- Binomial B(n,p) requires n fixed independent trials with constant p; normal calculations use z = (x − μ)/σ, with a continuity correction when a discrete variable is approximated.
- A hypothesis test compares the test statistic with the critical region or p-value at the stated significance level; the conclusion is in context.
Worked example
The mass X of an apple is modelled by X ~ N(120, 15²). Find P(110 < X < 140). Standardise: z₁ = (110 − 120)/15 = −0.667, z₂ = (140 − 120)/15 = 1.333. P = Φ(1.333) − Φ(−0.667) = 0.9088 − 0.2514 = 0.6574. So about 65.7% of apples have masses between 110 g and 140 g.
Marks that go missing
- Using class width instead of frequency density, or reading cumulative frequency at the wrong boundary.
- Applying the multiplication rule to events that are not independent, or the addition rule without subtracting the overlap.
- Forgetting the continuity correction when using a normal approximation.
- Using the sample correlation coefficient r as the gradient of the regression line, or extrapolating beyond the data.
- Choosing a one-tail test when the alternative is two-sided, or stating a conclusion without referring to the original context.