Probability and Statistics
Core concepts
Know classical probability, conditional probability, independence, and the addition and multiplication rules. For statistics, understand mean, median, mode, range, variance, standard deviation, and how to read distributions.
Traps
- Mutually exclusive vs independent: mutually exclusive events cannot occur together; independent events do not affect each other's probability.
- Conditional probability: P(A|B) = P(A ∩ B)/P(B), not P(A) in general.
- Mean vs median: outliers pull the mean but not the median.
- Standard deviation is the square root of variance; do not confuse the two.
- Sampling without replacement changes probabilities on each draw.
Rule of thumb
P(A or B) = P(A) + P(B) − P(A and B); drop the last term only if mutually exclusive.
Worked example
A bag has 4 red and 6 blue marbles. Draw two without replacement. P(both red) = (4/10)(3/9) = 12/90 = 2/15. P(at least one red) = 1 − P(no red) = 1 − (6/10)(5/9) = 1 − 30/90 = 2/3.
Mathematical Analysis and Calculus
Core concepts
Understand limits, continuity, derivatives (power, product, quotient, chain rules), applications of derivatives (tangent lines, monotonicity, extrema, optimization), and basic integration (antiderivatives, definite integrals, area under a curve).
Traps
- Chain rule omissions: d/dx[f(g(x))] = f'(g(x))·g'(x); the inner derivative is easy to forget.
- Product vs quotient signs: practice the formulas until automatic.
- Critical points vs endpoints: extrema on a closed interval occur at critical points or endpoints.
- Definite integral sign: reversing limits flips the sign.
- Constant of integration: include + C for indefinite integrals.
Rule of thumb
For optimization, set f'(x) = 0, solve, then use the second derivative (or a sign chart) to classify each critical point.
Worked example
Find the tangent line to f(x) = x³ − 3x at x = 1. Then f(1) = 1 − 3 = −2. Differentiate: f'(x) = 3x² − 3, so f'(1) = 0. The tangent is horizontal: y = −2.