Probability and Statistics: Counting, Distributions, and Interpretation
Core concepts
This domain blends counting techniques with probability rules and statistical reasoning. You should be able to apply the fundamental counting principle, permutations (order matters, (nPr = \frac{n!}{(n-r)!})), and combinations (order does not matter, (nCr = \frac{n!}{r!(n-r)!})). Probability questions often use these counts in the classical definition (P = \frac{\text{favourable}}{\text{total}}), or combine them with the addition rule (P(A \cup B) = P(A)+P(B)-P(A \cap B)) and the multiplication rule for independent events.
Expect binomial settings: a fixed number of independent trials, two outcomes, constant probability. Then (P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}). For statistics, understand measures of central tendency and spread, how outliers affect mean versus median, and how to read normal-distribution problems using z-scores and the empirical rule (68–95–99.7).
Traps to watch
- Using permutations when order is irrelevant (choosing a committee) or combinations when order matters (arranging a password).
- Double counting in "at least one" problems; the complement (1-P(\text{none})) is usually faster.
- Assuming events are independent without justification.
- Confusing standard deviation with variance, or reporting mean when data are skewed.
Rule of thumb
Ask two questions before computing: does order matter, and are events independent? The answers select the formula.
Worked example
A bag has 5 red and 3 blue marbles; draw 2 without replacement. (P(\text{both red}) = \frac{5}{8}\cdot\frac{4}{7} = \frac{20}{56} \approx 0.357).