A-Level Maths: Probability and Distributions

Probability and distributions form the computational core of A-Level statistics: conditional reasoning, the binomial model and the Normal model, each with conditions that must be checked before the formulas apply. This quiz covers all three layers.

The probability questions start with the without-replacement classic — the conditional probability that a second marble is blue given a red first draw, where the denominator must shrink to 7 — and the independence rules: P(A ∩ B) as the product and P(A|B) collapsing to P(A) when events are genuinely independent, tested with concrete values 0.4 and 0.3.

Discrete random variables appear through expectation: computing E(X) from a three-value distribution table, and the linearity properties of expected value tested as statements. The transformation question — mean 10, variance 4, Y = 2X + 3 — requires both rules at once: the constant shifts the mean but not the spread, and the factor of 2 scales the standard deviation to 4. The verification pass confirmed both keyed answers independently.

The binomial questions cover conditions and computation: which requirements genuinely define B(n, p) — fixed trials, independence, constant probability — and the full probability calculation P(X = 2) for B(10, 0.2), with binomial coefficient, success and failure powers all in play.

The Normal questions build the standardising workflow: computing a z-value from μ = 50, σ = 5, translating P(X > 108) for N(100, 16) into P(Z > 2) — where reading σ = 4 from the variance 16 is the tested step — and the standard Normal's symmetry properties, the facts that make table-free reasoning possible.

Explanations show every calculation in full, and distractors encode the classic confusions: variance for standard deviation, unchanged denominators after removal without replacement, and shifted spreads.

• Compute conditional probabilities in without-replacement settings
• Apply independence rules for intersections and conditional probabilities
• Calculate expected values and transform means and standard deviations linearly
• Verify binomial conditions and compute binomial probabilities
• Standardise Normal variables and use symmetry of the standard Normal

Topic scope follows sections M–N (Probability; Statistical distributions) of the DfE's prescribed AS and A level mathematics subject content — 100% common across all exam boards: conditional probability and independence, discrete random variables, the binomial distribution, and the Normal distribution.