Page 1 · Introduction
What is probability?
Probability measures how likely something is to happen. It sounds complicated, but here's a secret that will carry you through this whole module: almost every probability question comes down to careful counting. If you can count, you can do this.
Every probability is a number from 0 to 1:
- 0 — it will never happen (rolling a 7 on a normal dice).
- 1 — it will always happen (rolling a number under 7).
- In between — written as a fraction, decimal or percentage.
The big ideaProbability = favourable outcomes ÷ total outcomes. Count the ways you want, count the ways in total, divide. That's it.
💡Don't worry if it feels new — we'll build up slowly, from easy to challenging, one idea at a time. By the end you'll handle questions that stump most students.
On the real testWhere this appearsProbability sits in the Mathematical Reasoning section of the NSW Selective test (25% of your mark, computer-based from 2025). Quick, accurate counting under time pressure is exactly what's rewarded.
Page 2 · The language of chance
Putting events on a scale
Before we calculate, let's get a feel for likelihood. Every event sits somewhere on this scale:
Impossible (0) — Unlikely — Even chance (½) — Likely — Certain (1)
- Even chance (½) — a coin landing heads.
- Unlikely — rolling three sixes in a row.
- Likely — rolling more than 1 on a dice.
Why this matters
The test sometimes asks you to order events by likelihood, or match an event to a word. Knowing this scale means you can answer those instantly — and it gives you a sanity check: if you calculate a "likely" event and get a tiny number, you know to look again.
✅Quick self-check: is drawing a red card from a normal deck impossible, even chance, or certain? (Even chance — half the deck is red.)
Page 3 · The core method
The three-step method
Here is the method you'll use on every single question. Learn these three steps and you're set.
P(event) = favourable outcomes ÷ total outcomes
Step 1 — Count the total
How many outcomes are possible altogether? A coin = 2. A dice = 6. A bag of 10 marbles = 10.
Step 2 — Count the favourable
How many outcomes are the ones you want? Rolling an even number on a dice = 2, 4, 6 = three favourable outcomes.
Step 3 — Divide and simplify
P(even) = 3 ÷ 6 = ½. Always simplify your fraction at the end.
💡Say the three steps out loud as you work: "total… favourable… divide." Making it a habit means you'll never freeze on a question.
Page 4 · Worked examples
Let's walk through some together
Notice how each one uses the same three steps — total, favourable, divide.
1. A bag has 4 red, 3 blue and 5 green marbles. P(blue)?
Total: 4+3+5 = 12. Favourable: 3 blue. Divide: 3/12 = ¼.
Answer: ¼ (25%)
2. A fair dice is rolled. P(a number greater than 4)?
Total: 6. Favourable: 5 and 6 = 2. Divide: 2/6 = ⅓.
Answer: ⅓
3. A letter is picked from MERIT. P(a vowel)?
Total: 5 letters. Favourable: E, I = 2 vowels. Divide: 2/5.
Answer: ⅖
✅See the pattern? Every question — no matter how it's dressed up — is just those three steps.
Page 5 · A powerful shortcut
The "NOT" trick
The test loves asking for the chance something doesn't happen. Counting all those outcomes can be slow — so use this shortcut:
P(not happening) = 1 − P(happening)
Why does it work? Because something either happens or it doesn't — those two chances always add up to 1 (a certainty).
The chance of rain is ¼. Chance of no rain?
1 − ¼ = ¾.
Answer: ¾
When to reach for it
If a question says "at least one", "not", or "fewer than", the shortcut is usually faster than counting directly. Keep it in your back pocket — it turns hard questions into one subtraction.
Page 6 · Getting the answer to match
Simplifying fractions
The test's multiple-choice options are almost always in simplest form. If your answer is 4/12 but you see ⅓ in the options, they're the same — you just need to simplify.
How to simplify
Divide top and bottom by the same number until you can't go further:
- 4/12 → divide by 4 → ⅓
- 6/9 → divide by 3 → ⅔
- 10/100 → divide by 10 → 1/10
💡Stuck because your answer isn't in the options? Nine times out of ten it just needs simplifying. Don't panic — simplify and look again.
Page 7 · Practice
Level 1 · Easy
Warm up
Gentle ones to build confidence. Tap an answer to check it.
1. A spinner numbered 1–8. P(a number less than 3)?
Less than 3 = 1, 2 = 2 favourable, out of 8. 2/8 = ¼.
2. A bag has 6 red and 9 yellow counters. P(red)?
Total 6+9 = 15, red = 6. 6/15 = ⅖.
3. A fair dice is rolled. P(an even number)?
Even = 2, 4, 6 = 3 out of 6 = ½.
🎉Got these? Brilliant — you've got the core method down. Let's step it up.
Page 8 · Practice
Level 2 · Medium
Building up
These add a small twist — a "not", or working from a probability back to a count.
1. A spinner numbered 1–12. P(a multiple of 4)?
Multiples of 4: 4, 8, 12 = 3 out of 12 = ¼.
2. In a class of 30, 18 are girls. P(picking a boy)?
Boys = 30−18 = 12. 12/30 = ⅖. (Or use the shortcut: 1 − 18/30 = ⅖.)
3. A bag of 20 counters has P(green) = ⅖. How many are green?
Work backwards: ⅖ of 20 = 8. Multiply the probability by the total.
💪Notice these needed a second step. If one tripped you, re-read its explanation — that's how the tough ones start to click.
Page 9 · Practice
Level 3 · Hard
Real exam difficulty
These are the kind that separate strong students. Take your time — the method still works.
1. Two fair dice are rolled. P(the two add up to 9)?
Total = 6×6 = 36. Ways to make 9: (3,6)(4,5)(5,4)(6,3) = 4. So 4/36 = 1/9. Key insight: (3,6) and (6,3) are different outcomes.
2. A number is picked from 1–20. P(NOT a multiple of 3)?
Multiples of 3 up to 20: 3,6,9,12,15,18 = 6. Shortcut: 1 − 6/20 = 14/20 = 7/10.
3. A bag has 5 red and 3 blue. How many extra red are added so P(red) = ¾?
The 3 blue never change and must be ¼ of the total, so total = 12, meaning 9 red. Started with 5, so add 4. Working from the unchanging part is the smart move.
💡These are genuinely hard — if you got even one, you're thinking like a top student. The explanations show the shortcut for next time.
Page 10 · Challenge
Level 4 · Challenge
Think smart
The toughest tier. Each needs a clever insight, not just the formula. Don't be discouraged if these take a few tries.
1. A coin is flipped twice. P(at least one head)?
"At least one" is hard to count directly, but its opposite — "no heads" (TT) — is easy: ¼. So 1 − ¼ = ¾. Flipping to the opposite is a top exam skill.
2. A bag has red and blue in ratio 3:2. After 10 red are removed, it's 1:2. How many blue?
Blue never changes — that's the key. Blue = 2 units throughout. Red goes 3 units → 1 unit, a drop of 2 units = 10 removed, so 1 unit = 5. Blue = 2×5 = 10.
3. A spinner numbered 1–10 is spun twice. P(both spins are even)?
P(even) each spin = 5/10 = ½. For BOTH, multiply: ½ × ½ = ¼. Independent events multiply — a powerful idea for the hardest questions.
🌟These are beyond what most students can do. Every attempt makes you sharper — be proud of sticking with them.
Complete
Outstanding work!
You've completed the Probability module.
You've gone from the basics all the way to challenge-level questions. You now know the three-step method, the "not" shortcut, simplifying, working backwards, and even multiplying independent events. That's a genuinely strong toolkit — keep practising and it becomes automatic.