Statistics and Probability
What this domain tests
This statistics and probability chapter explores how data are collected, summarized, and interpreted, and how probability models uncertainty. Expect measures of centre and spread, graphical displays, normal distributions and z-scores, counting-based probability, conditional probability, and critical evaluation of study design and correlation versus causation.
Core rules and concepts
- Centre and spread: mean is sensitive to outliers, median is resistant; standard deviation measures typical distance from the mean. Range ignores the middle of the data.
- Probability rules: 0 ≤ P(A) ≤ 1; P(A′) = 1 − P(A); P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for independent events P(A ∩ B) = P(A)P(B).
- Conditional probability: P(A|B) = P(A ∩ B)/P(B).
- Counting: permutations count ordered arrangements (nPr), combinations count unordered selections (nCr).
- Normal distribution: z = (x − μ)/σ standardizes values; roughly 68% of data lie within one standard deviation, 95% within two.
- Study design: random sampling supports generalization; random assignment supports causal claims; observational association does not establish causation.
Worked example
A test has 10 multiple-choice questions, each with 4 options. A student guesses every answer. What is the probability of exactly 3 correct?
This is binomial: n = 10, p = 1/4, k = 3. The number of ways to choose which 3 are correct is C(10,3) = 120. Then: P = 120 × (1/4)³ × (3/4)⁷ = 120 × (1/64) × (2187/16384) = 120 × 2187 / 1,048,576 ≈ 0.2503
So the probability is about 25%. Note the exponent 7 on the failure probability - the remaining questions must be wrong, not merely unspecified.
Traps that cost marks
- Using permutations where order does not matter (overcounting combinations).
- Treating dependent events as independent, or omitting the overlap term in P(A ∪ B).
- Reversing conditional probability: P(A|B) is not P(B|A).
- Reading a z-score table in the wrong direction or forgetting to subtract from 1 for upper tails.
- Claiming causation from an observational correlation.
- Using mean as a summary when the distribution is strongly skewed.