Data Analysis, Statistics, Probability, and Discrete Mathematics
Data, probability, and discrete structures
Candidates must summarize data using mean, median, mode, range, interquartile range, and standard deviation; interpret histograms, boxplots, and scatterplots; compute and interpret z-scores and the empirical rule; and recognize sampling bias. For probability, know sample spaces, addition and multiplication rules, conditional probability, independence, expected value, and complements. Discrete topics include counting principles, permutations and combinations, binomial probability, recursion, sets, logic, and basic graph and network ideas.
Traps
- Mutually exclusive is not the same as independent. Independent events satisfy P(A∩B) = P(A)P(B); mutually exclusive events have P(A∩B) = 0.
- P(A|B) is not P(B|A); use P(A|B) = P(A∩B)/P(B).
- "At least one" is usually easiest through 1 − P(none).
- Sampling without replacement changes denominators and can destroy independence.
- Correlation does not imply causation, and extrapolation beyond the data is risky.
Worked example
A fair coin is flipped three times. P(at least one head) = 1 − P(no heads) = 1 − (1/2)³ = 7/8. Rule of thumb: when a question says "at least one," compute the complement first.