Probability and Statistics
The Probability and Statistics domain tests counting, the laws of probability including conditional probability and independence, and the standard distributions with their means and variances. Questions are usually set in context - counters drawn from a bag, items from a production line - and require both a computed probability and an interpretation of it.
Core rules
- P(A ∪ B) = P(A) + P(B) - P(A ∩ B); P(A') = 1 - P(A).
- P(A | B) = P(A ∩ B)/P(B); A and B are independent exactly when P(A ∩ B) = P(A)P(B).
- Selections use C(n, r) = n!/(r!(n - r)!); arrangements use P(n, r) = n!/(n - r)!.
- For a random variable X: E(aX + b) = aE(X) + b and Var(aX + b) = a^2 Var(X); the standard deviation is the square root of the variance.
- Binomial B(n, p): P(X = r) = C(n, r) p^r (1 - p)^(n - r), with mean np and variance np(1 - p).
- Normal N(μ, σ^2): standardise with z = (x - μ)/σ and use the table; apply a continuity correction when approximating a discrete distribution.
Worked example
A fair die is rolled five times and X counts the sixes, so X ~ B(5, 1/6). Then P(X ≥ 1) = 1 - P(X = 0) = 1 - (5/6)^5 ≈ 1 - 0.4019 = 0.598.
Traps
- Treating sampling without replacement as binomial, where the trials are not independent.
- Writing Var(2X) = 2Var(X); the correct value is 4Var(X).
- Omitting the continuity correction when a normal approximation to a binomial is used.
- Counting ordered arrangements when the question asks for a selection, or ignoring identical items.