Probability and Statistics
The Probability and Statistics domain covers probability rules, counting, distributions, descriptive statistics, and the logic of inference.
Probability rules
0 ≤ P(A) ≤ 1, P(Aᶜ) = 1 − P(A), and P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Events are independent when P(A ∩ B) = P(A)P(B); mutually exclusive events have P(A ∩ B) = 0 and cannot be independent unless one has probability zero. Conditional probability: P(A|B) = P(A ∩ B)/P(B), rearranged into the multiplication rule and extended by Bayes' theorem.
Counting
Permutations count ordered arrangements: nPr = n!/(n−r)!. Combinations count unordered selections: nCr = n!/(r!(n−r)!).
Statistics
Mean, median, and standard deviation summarize center and spread; z = (x − μ)/σ standardizes. The empirical rule gives roughly 68%, 95%, and 99.7% within 1, 2, and 3 standard deviations for normal data. The Central Limit Theorem describes the sampling distribution of the mean. A p-value is the probability of data at least as extreme, assuming the null hypothesis is true - not the probability that the null is true.
Worked example
Two cards are drawn without replacement from a standard 52-card deck. P(both aces) = (4/52)(3/51) = 12/2652 = 1/221 ≈ 0.0045. The second factor changes because the first draw removed an ace.
Traps
Confusing P(A|B) with P(B|A); treating dependent draws as independent; computing at least one directly instead of using 1 − P(none); misreading a p-value as evidence about the null.