Statistics and Probability
Core concepts
Statistics summarise data: mean (average), median (middle value), mode (most frequent), and range (max − min). The mean is sensitive to outliers; the median is resistant. For probability, P(event) = favourable outcomes / total equally likely outcomes, always between 0 and 1. Complementary events satisfy P(not A) = 1 − P(A). For independent events, multiply: P(A and B) = P(A)·P(B). For mutually exclusive events, add: P(A or B) = P(A) + P(B). Understand sample space (all possible outcomes) and how to count it with tables or tree diagrams. Recognise sampling ideas: a random sample better represents a population than a biased or convenience sample. Interpret data displays - mean, median, and spread - to compare two data sets.
Traps to expect
- Adding probabilities for events that are not mutually exclusive (double-counting the overlap).
- Multiplying probabilities for dependent events without adjusting the second probability.
- Confusing mean and median when an outlier skews the data.
- Forgetting that probabilities must sum to 1 across all outcomes.
- Assuming a small or self-selected sample represents the whole population.
Worked example
A bag has 3 red and 5 blue marbles (8 total). P(red) = 3/8. P(not red) = 1 − 3/8 = 5/8. Drawing twice with replacement, P(red then blue) = (3/8)(5/8) = 15/64, because the draws are independent.
Rule of thumb
Ask whether events are independent (multiply) or mutually exclusive (add), and check that your probability lies between 0 and 1.
Financial and Discrete Mathematics
Core concepts
Financial mathematics applies percentages and formulas to money. Understand percent change: new = old × (1 ± r). Simple interest is I = Prt (principal × rate × time). Compound interest is A = P(1 + r/n)^(nt), where n is compounding periods per year. Know how to compute markups, discounts, tax, and tips, and how to work backward from a final price to the original. Discrete mathematics covers counting and logical structures: the fundamental counting principle (multiply choices across stages), permutations (order matters) versus combinations (order does not), and simple sequences and patterns. Read pay structures (hourly, salary, overtime) and budgets, distinguishing gross pay from net pay after deductions.
Traps to expect
- Applying a discount and then a tax in the wrong order, or confusing percent of the original with percent of the new amount.
- Using simple interest when the problem states compounding (or vice versa).
- Forgetting to convert a percent to a decimal and time to years.
- Treating a permutation as a combination when order matters (e.g. rankings vs. committees).
- Confusing gross pay (before deductions) with net pay (take-home).
Worked example
Invest $1,000 at 6% compounded annually for 2 years. A = 1000(1 + 0.06)² = 1000(1.1236) = $1,123.60. Simple interest would give 1000 + 1000(0.06)(2) = $1,120, so compounding earns slightly more.
Rule of thumb
Write the formula first, convert all percents to decimals and time to years, then substitute - and check whether the problem says simple or compound.