Combinatorics, Probability, and Statistics
Core concepts
Know the multiplication and addition principles, permutations (ordered), combinations (unordered), permutations with repetition, and the binomial theorem. For probability, master classical probability, complementary events, mutually exclusive events, independent events, and conditional probability. For statistics, know mean, median, mode, variance, standard deviation, and how outliers and shifts affect each.
Traps
- Using combinations when order matters, or permutations when it does not.
- Forgetting to subtract overcounted cases or to divide by repeated factorials for identical items.
- Assuming independence when it is not stated; P(A ∩ B) = P(A)P(B) only for independent events.
- Adding probabilities of events that are not mutually exclusive without subtracting the intersection.
- Believing the median changes when a large outlier is added; the mean does, the median often does not.
Worked example
A committee of 3 is chosen from 5 men and 4 women. The probability it contains exactly 2 women: total committees C(9,3) = 84; favorable C(4,2)·C(5,1) = 6·5 = 30. Probability = 30/84 = 5/14.
Rule of thumb
Ask whether order matters before choosing between P(n,k) and C(n,k); for "at least one" problems use the complement 1 − P(none).