Vu's Maths Hub: HSC Probability: Conditional Events, Independence, and Binomial Models

Intro

The HSC Probability booklet is a free Mathematics Advanced and Extension 1 resource with 67 worked problems covering sample spaces, conditional probability, independence, counting-based probability, discrete random variables, and binomial distributions. It is written for Extension 1 students wanting structured probability practice; tutors and teachers and designed for structured HSC revision on Vu's Maths Hub.

This deep-dive introduces HSC Probability — browser-readable, aligned with the NESA syllabus.

Summary

The HSC Probability booklet offers a fundamentals review, 67 tiered problems with solutions, appendices, and a conclusion that distils exam habits. Open HSC Probability — no account required. Use this post to plan how to work through a 74-page booklet efficiently.

What is this booklet?

This booklet is written for Extension 1 students wanting structured probability practice; tutors and teachers.

Focus: sample spaces, conditional probability, independence, counting-based probability, discrete random variables, and binomial distributions.

Topics covered:

Sample spaces
Events and set operations
Addition and multiplication rules
Conditional probability and independence
Counting-based probability
Discrete random variables
Expected value and variance
Binomial distributions

How to use it:

Read fundamentals first
Part 1 for model HSC reasoning; Part 2 for fluency
Watch wording: at least, exactly, given that, independent
Distinguish events from random variables

Approximately 74 pages, CC BY 4.0, readable at HSC Probability.

Key fundamental reviews

The booklet opens with a dedicated Fundamentals Review — read it before Part 1. It is a compact reference for notation, formulas, and reasoning patterns:

Events, complements, unions, and intersections
Addition rule and mutually exclusive cases
Conditional probability $P (A ∣ B)$ and multiplication rule
Independence vs dependence
Counting methods linked to probability
Expected value and variance; binomial conditions

Why fundamentals matter

Events, complements, unions, and intersections — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Addition rule and mutually exclusive cases — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Conditional probability $P — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Independence vs dependence — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Counting methods linked to probability — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Students who skip this section often repeat the same algebra errors in Part 2. Treat HSC Probability fundamentals as a closed-book quiz first.

Problems and how to use them

The HSC Probability booklet packs 67 practice problems into roughly 74 pages — well beyond a single textbook chapter. Each item includes worked solutions; many include Takeaways that highlight the method to reuse in exams.

Overall structure

Part 1 — detailed solutions: basic, medium, and advanced probability with model HSC reasoning.

Part 2 — hint-based fluency: fluency drills with upside-down hints across three tiers.

Use Part 1 to learn how complete NSW HSC working is written. Use Part 2 in the fortnight before trials — hints are upside-down so you attempt first.

Part 1 (20 problems)

Advanced (7 problems)

This tier contains 7 problems aimed at extension and synthesis. Representative work includes "A probability table with unknown constant" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Advanced and Extension 1 marking expectations.

A probability table with unknown constant
Exactly two successes in a binomial setting
At least one defective item
A binomial distribution from a target score
Comparing two games by expectation
HSC 2020-style: Biased coin and normal approximation
HSC 2022-style: When a normal approximation may fail

Basic (4 problems)

This tier contains 4 problems aimed at foundational fluency. Representative work includes "Two dice and a sum of at least 10" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Advanced and Extension 1 marking expectations.

Two dice and a sum of at least 10
A card that is a heart or a face card
At least one blue marble
Mutually exclusive or independent?

Medium (9 problems)

This tier contains 9 problems aimed at exam-standard reasoning. Representative work includes "Conditional probability from a class survey" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Advanced and Extension 1 marking expectations.

Conditional probability from a class survey
Checking independence
Why independence does not depend on order
Expected score of a game
A fair n-sided spinner
A random sum from a multiset
Selection probability via combinations
Leading digit of a height measurement
HSC 2023-style: Sample proportion and normal approximation

Part 2 (47 problems)

Advanced (17 problems)

This tier contains 17 problems aimed at extension and synthesis. Representative work includes "No sixes in repeated rolls" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Advanced and Extension 1 marking expectations.

No sixes in repeated rolls
At most two successes
Game value
Variance from a small distribution
Exactly three correct guesses
Expected number of red balls
A fair game check
More than half successes
Conditional probability after a draw
Geometric probability in a rectangular field
Coin on a square grid
HSC 2024-style: Sample size from standard deviation
HSC 2024-style: Charity donations
Designing the ``Lucky 3'' lottery game
The truncated St Petersburg game
HSC 2025-style: Inverse-tail green-counter problem
HSC 2022-style: Airline overbooking threshold

Basic (13 problems)

This tier contains 13 problems aimed at foundational fluency. Representative work includes "One card from a deck" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Advanced and Extension 1 marking expectations.

One card from a deck
Three coin tosses
Without replacement
Expected value from a table
Hannah and the snack boxes
Reading probabilities from a table
Complement of an impossible pair
Single-stage conditional probability
Binomial single success
One coin and one die
No red counter
A letter from MIRANDA
HSC 2025-style: Bernoulli mean and variance

Medium (17 problems)

This tier contains 17 problems aimed at exam-standard reasoning. Representative work includes "Conditional selection" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Advanced and Extension 1 marking expectations.

Conditional selection
Independence check
Finding a missing probability
Binomial mean and variance
A union from a survey table
Variance from a probability table
Distribution table from draws without replacement
A first binomial distribution table
A first hypergeometric distribution table
At least two heads
Exactly one black card
Expected number of tails
Fair coin tossed n times
HSC 2024-style: Driver test
HSC 2024-style: Same-metal coins
HSC 2023-style: Mixed equipment probabilities
HSC 2021-style: A Bernoulli complement

Common patterns across the booklet

Complex numbers: 1 problem — e.g. "A binomial distribution from a target score"
Probability & counting: 19 problems — e.g. "A probability table with unknown constant"
Trigonometry: 1 problem — e.g. "HSC 2024-style: Sample size from standard deviation"
Sequences & series: 1 problem — e.g. "A first hypergeometric distribution table"
Other synthesis: 45 problems — e.g. "At least one defective item"

Standout and less-seen problem types

These go beyond routine drills — expect unfamiliar wording or multi-topic synthesis:

A probability table with unknown constant: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
Exactly two successes in a binomial setting: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
At least one defective item: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
A binomial distribution from a target score: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
Comparing two games by expectation: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
HSC 2020-style: Biased coin and normal approximation: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.

Working through a large booklet

Read fundamentals first
Part 1 for model HSC reasoning; Part 2 for fluency
Watch wording: at least, exactly, given that, independent
Distinguish events from random variables

Timed practice: allow about one minute per mark; write legible structure even when practising alone. Error log: tag mistakes as concept, algebra, or misreading. Rotation: do not camp on advanced tier if basics still slip.

Open HSC Probability and work steadily — 67 problems is a marathon, not a sprint.

Key appendices

Formula sheet — Rules, notation, and binomial summary. Use for quick reference or enrichment beyond routine exam questions.

Common probability traps — Wording pitfalls and double-counting. Use for quick reference or enrichment beyond routine exam questions.

Key conclusion

Define events clearly; use complements; distinguish conditional from ordinary probability; treat expectation as a weighted average; recognise binomial structure only when conditions hold.

The booklet's closing section reinforces these habits:

Read fundamentals first
Part 1 for model HSC reasoning; Part 2 for fluency
Watch wording: at least, exactly, given that, independent
Distinguish events from random variables

Revisit HSC Probability in the fortnight before trials and redo problems you missed on first pass.

How to study with this booklet

Fundamentals plus Part 1 basic in week 1; add Combinatorics for counting setups; finish with Distributions for random-variable questions

General principles:

Closed-book first: attempt without notes, then check fundamentals.
Error log: record concept vs algebra vs reading errors.
Spaced repetition: redo missed questions after 3 and 7 days.
Past papers last: fix weak topics here, then sit full papers timed.

Mini-FAQ

Who is the HSC Probability booklet for?

Extension 1 students wanting structured probability practice; tutors and teachers studying Mathematics Advanced and Extension 1 under the NSW HSC.

Should I read solutions before attempting problems?

Attempt Part 1 first. Use Part 2 hints only after a genuine try or partial working.

Where can I read the booklet online?

Open HSC Probability on Vu's Maths Hub — free, no account required.

How many problems are in the booklet?

Roughly 67 practice problems across 74 pages, each with worked solutions.

Is this aligned with NESA?

Topics match Mathematics Advanced and Extension 1 outcomes for sample spaces, conditional probability, independence, counting-based probability, discrete random variables, and binomial distributions. Confirm scope with your teacher and current NESA documentation.

Common mistakes to avoid

Assuming independence without justification
Using $P (A \cap B) = P (A) P (B)$ when events are conditional
Misreading "at least one" — complement is often easier
Applying binomial when trials are not identical or independent
Rushing to advanced tiers before basic fluency — build foundations first.

Practice on Vu's Maths Hub

Open the free HSC Probability on Vu's Maths Hub — 67 problems with full worked solutions.

Related resources:

How to use Vu's Maths Hub — Probability revision workflow
HSC Combinatorics — Counting for probability

Sample spaces — exam context

In NSW Mathematics Advanced and Extension 1 examinations, sample spaces routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Probability booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Probability and locate items that stress sample spaces; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Events and set operations — exam context

In NSW Mathematics Advanced and Extension 1 examinations, events and set operations routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Probability booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Probability and locate items that stress events and set operations; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Addition and multiplication rules — exam context

In NSW Mathematics Advanced and Extension 1 examinations, addition and multiplication rules routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Probability booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Probability and locate items that stress addition and multiplication rules; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Conditional probability and independence — exam context

In NSW Mathematics Advanced and Extension 1 examinations, conditional probability and independence routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Probability booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Probability and locate items that stress conditional probability and independence; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Counting-based probability — exam context

In NSW Mathematics Advanced and Extension 1 examinations, counting-based probability routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Probability booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Probability and locate items that stress counting-based probability; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Discrete random variables — exam context

In NSW Mathematics Advanced and Extension 1 examinations, discrete random variables routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Probability booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Probability and locate items that stress discrete random variables; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Expected value and variance — exam context

In NSW Mathematics Advanced and Extension 1 examinations, expected value and variance routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Probability booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Probability and locate items that stress expected value and variance; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Binomial distributions — exam context

In NSW Mathematics Advanced and Extension 1 examinations, binomial distributions routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Probability booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Probability and locate items that stress binomial distributions; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Syllabus alignment

This booklet supports Mathematics Advanced and Extension 1 under the NESA syllabus. It supplements school instruction with 74 pages of extra exam-style practice — not a replacement for class teaching.

Additional exam advice

When sitting Mathematics Advanced and Extension 1 exams, allocate time proportional to marks. Practise concise justification in HSC Probability — NSW markers reward clear communication. Reread the booklet conclusion the night before for a habit checklist.

Why Vu's Maths Hub

Vu's Maths Hub hosts every HSC booklet in a continuous, mobile-friendly viewer — zoom, search, no download required. Maintained by Vu Hung Nguyen; CC BY 4.0 for personal and school use.

Wednesday, 1 July 2026

HSC Probability: Conditional Events, Independence, and Binomial Models