Probability and statistics
What this domain tests
Probability rules, conditional probability, independence, discrete random variables and descriptive statistics. Later questions often require a clear tree diagram or two-way table before any calculation.
Core rules
- Addition rule: P(A union B) = P(A) + P(B) - P(A intersection B). For mutually exclusive events the last term is zero.
- Conditional probability: P(A|B) = P(A intersection B)/P(B), provided P(B) > 0.
- Independence: A and B are independent exactly when P(A intersection B) = P(A)P(B), equivalently P(A|B) = P(A).
- Total probability and Bayes: P(B) = sum over A_i of P(B|A_i)P(A_i); then P(A|B) = P(B|A)P(A)/P(B).
- Binomial distribution: P(X = k) = C(n, k) p^k (1 - p)^(n-k), with E(X) = np and Var(X) = np(1 - p).
- Statistics: mean, median and mode measure centre; range, interquartile range and standard deviation measure spread. The standard deviation is the square root of the mean of the squared deviations.
Worked example
A disease affects 1 percent of a population. A test detects it in 95 percent of cases and gives a false positive in 5 percent of healthy people. Given a positive test, what is the probability of disease? Let D be disease and + a positive test. P(D) = 0.01, P(+|D) = 0.95, P(+|not D) = 0.05. Then P(+) = 0.01(0.95) + 0.99(0.05) = 0.0095 + 0.0495 = 0.059. Therefore P(D|+) = 0.0095/0.059, approximately 0.161. Despite the accurate test, only about 16 percent of positive results are true positives, because the disease is rare.
Traps that cost marks
- Confusing P(A|B) with P(B|A), the classic inversion error.
- Assuming independence because events look unrelated, without checking the product rule.
- For at-least-one questions, failing to use the complement 1 - P(none).
- Choosing the mean for skewed data when the median is the better measure.