HSC Probability: Study Guide & Outline

The introduction sets the scene by describing the booklet's purpose as a targeted resource for HSC probability. It identifies the target audience (primarily Advanced and Extension 1 students), explains how to navigate the worked examples and practice banks, and previews the topics covered—from fundamentals such as set notation and basic rules through to advanced distributions and sampling. The section closes with a note on how the concepts lead into further study, providing motivation and a roadmap.

This chapter reviews all essential probability concepts required for HSC. It begins with core language and set notation refreshers, then formalises complements, unions, intersections, mutually exclusive events, the multiplication rule, and conditional probability. Independence is defined and linked to counting methods, tree diagrams, two-way tables, and staged experiments. The second half introduces discrete random variables, expectation ($\operatorname{E}(X) = \sum x\,p(x)$) and variance ($\operatorname{Var}(X) = \operatorname{E}(X^2) - [\operatorname{E}(X)]^2$), followed by the binomial distribution $X \sim \operatorname{Bin}(n,p)$ and the sample proportion $\hat{p} = X/n$. The normal approximation to the binomial ($\hat{p} \stackrel{\text{approx}}{\sim} N\big(p, \frac{p(1-p)}{n}\big)$) is discussed with conditions $np \ge 10$ and $n(1-p) \ge 10$. A method selection checklist and common language triggers (e.g., 'at least', 'given that', 'exactly') help students match problem statements to appropriate techniques.

Part 1 provides fully worked solutions to 20 problems organised by difficulty. Basic problems (e.g., dice sums, card draws, complementary probability) reinforce $P(A \cup B)$ and $P(A' )$ usage. Medium problems explore conditional probability from tables, independence checks, expected value of a game, and binomial probabilities involving combinations. Advanced problems extend to normal approximation for sample proportions, inverse questions (e.g., finding $n$ to achieve a desired standard deviation), and evaluating when normal approximation may fail. Each solution models systematic thinking: defining events, applying formulas, drawing diagrams, and checking assumptions. Key takeaways include interpreting 'at least one' as $1 - P(\text{none})$, verifying $np$ and $n(1-p)$ conditions, and using the complementary rule for binomial tails.

Part 2 is a practice bank of 47 problems with concise hints and solutions. Basic problems (e.g., one card, three coin tosses, without replacement, binomial single success) build fluency with core rules. Medium problems extend to conditional selection, independence checks, finding missing probabilities, binomial mean/variance, hypergeometric distributions, and expected numbers. Advanced problems cover no-sixes rolls, at most two successes, game fairness, geometric probability (coin on a grid, rectangular field), inverse binomial (finding sample size from standard deviation), charity donation modelling, designing a lottery game, a truncated St Petersburg game, and overbooking thresholds. This section reinforces techniques such as drawing tree diagrams, using two-way tables, applying the complement rule, and interpreting normal approximations in context.

The appendices provide a compact formula sheet covering probability notation, addition/multiplication rules, $\operatorname{E}(X)$, $\operatorname{Var}(X)$, binomial distribution pmf and parameters, normal approximation conditions, and sample proportion formulas. The common probability traps section highlights frequent errors such as confusing mutually exclusive with independent events, misapplying the addition rule, mishandling conditional probability direction, and ignoring sample size requirements for normal approximation.

The conclusion summarises the booklet's philosophy, encourages consistent practice, and reminds students to use the method selection checklist when approaching HSC probability questions. It reinforces the importance of understanding underpinning concepts rather than rote application, aiming to build confidence for examinations.