Probability and statistics
This domain tests counting, probability rules, conditional probability, independence, the binomial distribution and simple descriptive statistics. Questions often combine a counting part with a conditional or binomial part.
Counting and basic rules
Permutations nPr = n!/(n − r)! count ordered selections; combinations nCr = n!/(r!(n − r)!) count unordered ones. P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for mutually exclusive events the intersection term is zero. Use P(A') = 1 − P(A), and for "at least one" use the complement.
Conditional probability and independence
P(A|B) = P(A ∩ B)/P(B). Events are independent if P(A ∩ B) = P(A)P(B), equivalently P(A|B) = P(A). Tree diagrams multiply along branches and add across mutually exclusive branches.
Binomial distribution
For n independent trials with constant success probability p, P(X = r) = nCr p^r (1 − p)^(n − r), with E(X) = np and Var(X) = np(1 − p). Descriptive statistics use mean = Σfx/Σf and standard deviation = √(Σf(x − mean)²/Σf).
Worked example
A student guesses all 8 questions of a multiple-choice test, each with 4 options. Find P(exactly 3 correct) and the expected number correct. Here n = 8, p = 1/4, so P(X = 3) = 8C3 (1/4)³ (3/4)⁵ = 56 × 0.015625 × 0.2373 ≈ 0.208. E(X) = np = 8 × 1/4 = 2.
Traps
Using nPr when order does not matter; forgetting the denominator changes in sampling without replacement; confusing mutually exclusive with independent; using the binomial when p changes between trials; assuming the expected value must be an attainable count.