Mathematics - Algebra, Functions, Probability, Statistics and Data Interpretation
The mathematics section tests algebraic manipulation, functions, probability and statistics, plus interpretation of tables and graphs presented in a scientific context. The arithmetic is rarely hard; the marks are lost on domain restrictions, conditional probability and variance conventions.
Core rules
- Functions. Domain requires denominator ≠ 0, radicand ≥ 0 and log argument > 0. f∘g means apply g first. An inverse exists only where f is one-to-one.
- Logs and exponentials. log(ab) = log a + log b; log(a/b) = log a − log b; log aⁿ = n log a. For N = N₀e^{kt}, half-life t½ = ln2/k.
- Probability. P(A∪B) = P(A) + P(B) − P(A∩B); P(A|B) = P(A∩B)/P(B); independence means P(A∩B) = P(A)P(B). Binomial: C(n,k)pᵏ(1−p)^{n−k}.
- Statistics. Sample standard deviation uses n − 1; the mean is pulled by outliers, the median is not; approximately 68/95/99.7% of a normal distribution lies within 1/2/3 SD.
Worked example
A test has 99% sensitivity and 95% specificity; prevalence is 1%. Of 10 000 people, 100 are diseased, and 99 of them test positive. Of the 9900 healthy, 5% (495) test positive falsely. So P(disease | positive) = 99/(99 + 495) = 99/594 ≈ 0.167, about 17%. The base rate, not the test's accuracy alone, drives the answer.
Traps
Confusing P(A|B) with P(B|A); forgetting to convert specificity into a false-positive rate (1 − specificity); dividing by n instead of n − 1 for a sample SD; accepting a domain that makes a logarithm or square root undefined.