Statistics, Probability, and Discrete Mathematics
Statistics, Probability, and Discrete Mathematics
This domain tests data summary and interpretation, probability rules, counting methods, and discrete structures such as sequences, graphs, and recurrence. Expect questions on center and spread, displays, independence, conditional probability, permutations and combinations, and simple graph models.
Core rules and concepts
- Mean is sensitive to outliers; median and mode are resistant. Range, interquartile range (IQR), and standard deviation describe spread.
- In a box plot, the box spans Q1 to Q3, the median is inside, and whiskers extend to the minimum and maximum (or within 1.5 IQR in a modified box plot).
- Probability: P(A^c) = 1 - P(A); P(A or B) = P(A) + P(B) - P(A and B); for independent events, P(A and B) = P(A)P(B).
- Conditional probability: P(A|B) = P(A and B)/P(B).
- Permutations count ordered arrangements: P(n, r) = n!/(n - r)!. Combinations count unordered selections: C(n, r) = n!/(r!(n - r)!).
- In graph models, an Euler circuit requires every vertex to have even degree; a tree with n vertices has n - 1 edges and no cycles.
Worked example
A committee of 4 is chosen from 10 students. How many committees are possible? Does the answer change if the 4 students have distinct roles?
Because a committee is unordered, use combinations: C(10, 4) = (10 · 9 · 8 · 7)/(4 · 3 · 2 · 1) = 210. If the students have distinct roles, order matters, so use permutations: 10 · 9 · 8 · 7 = 5040. The wording “committee” signals unordered selection; “president, vice president, secretary, treasurer” signals ordered roles.
Traps that cost marks
- Using permutations when order does not matter, or combinations when it does.
- Treating events as independent when sampling is without replacement.
- Confusing mutually exclusive events with independent events.
- Reporting correlation as proof of causation.
- Reading a cumulative frequency graph as if it showed individual frequencies.