Lesson 18 ~30 min Unit 4 · Probability +85 XP

Sample Space and Tree Diagrams

Master listing all outcomes systematically — tree diagrams, grid method, and the multiplication principle that makes counting effortless.

Today's hook: Tossing two coins — is the sample space {HH, TT, HT}? Three outcomes? Actually NO! HT and TH are different outcomes. Listing outcomes correctly is the foundation of all probability — mess this up and every answer is wrong.

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Think First

warm-up

Before you read on — quickly: A restaurant offers 2 choices of main (chicken or fish) and 3 choices of side (chips, salad, or rice). How many different meal combinations are possible? Try to list them all.

Record your answer in your workbook.

The Big Idea

+5 XP

A sample space is the complete list of ALL possible outcomes of an experiment. Tree diagrams organise multi-stage outcomes systematically by branching at each stage. The multiplication principle lets you count total outcomes without listing them all.

For a two-stage experiment, draw a branch for each possibility at Stage 1, then from each of those, draw branches for Stage 2. Count the paths from left to right. The total number of outcomes = options at stage 1 × options at stage 2. This is the multiplication principle.

Total outcomes = stage 1 × stage 2 = 2 × 2 = 4

HT ≠ TH

Heads then Tails is different from Tails then Heads. Order matters in two-stage experiments.

Outcomes at the end

Read across the full tree from left to right. Outcomes are listed at the tips of branches.

Multiply, don't add

Total outcomes = stage 1 × stage 2 (not stage 1 + stage 2).

What You'll Master

objectives

Know

A sample space lists ALL possible outcomes
Tree diagrams have stages with branches
Total outcomes = product of choices at each stage

Understand

Why HT and TH are different outcomes
Why the multiplication principle works
When to use a tree diagram vs a grid table

Can Do

Draw and label a complete tree diagram for a two-stage experiment
List all outcomes from a tree diagram
Apply the multiplication principle to count outcomes

Words You Need

vocabulary

Sample spaceThe complete set of all possible outcomes of a probability experiment.

Tree diagramA branching diagram that shows all possible outcomes of a multi-stage experiment.

BranchA line in a tree diagram representing one possible outcome at a stage.

StageOne event or selection in a multi-step experiment (e.g. the first coin flip).

Multiplication principleTotal outcomes = options at stage 1 × options at stage 2 × ... for independent stages.

Array/grid methodA table with stage 1 outcomes in rows and stage 2 in columns — each cell is one combined outcome.

Spot the Trap

heads-up

Wrong: "Flipping two coins gives 3 outcomes: HH, TT, or one of each." This treats HT and TH as the same outcome.

Right: There are 4 equally likely outcomes: HH, HT, TH, TT. HT (heads first, then tails) is different from TH (tails first, then heads).

Wrong: Listing outcomes at intermediate branches instead of the final tips of the tree.

Right: Read the full path from start to end. Each complete path (left to right) is one outcome: e.g. H → T gives HT.

Systematic Listing

+5 XP

A systematic list ensures no outcome is missed. Fix the first stage and vary the second, then move to the next first-stage option. A grid/array is useful for two-stage experiments: rows = stage 1, columns = stage 2.

A spinner has outcomes {A, B} and a die has faces {1, 2, 3}. Grid method: fix A, pair with 1, 2, 3 giving A1, A2, A3. Then fix B, pair with 1, 2, 3 giving B1, B2, B3. Total: 6 outcomes. Verified by multiplication: 2 × 3 = 6.

Fix stage 1 → vary stage 2 → repeat

Fix and vary

Fix stage 1, vary stage 2. Then move to next stage 1 option. Never skip.

Grid is visual

Grids work best for two stages with few options. Trees work for three or more stages.

Verify with multiply

Count your outcomes and check with multiplication. If they disagree, you missed one.

Tree Diagrams

+5 XP

A tree diagram starts at a point (root), then branches for each possible outcome at Stage 1, then branches again for Stage 2. Each complete path from root to tip is one outcome. Label every branch. List outcomes at the tips.

Flipping a coin then rolling a die (1–3): Stage 1 has 2 branches (H, T). From each, Stage 2 has 3 branches (1, 2, 3). Total leaves = 6 outcomes: {H1, H2, H3, T1, T2, T3}. Multiplication: 2 × 3 = 6. We can find P(H1) = 1/6.

Label branches → read paths → list outcomes

Every branch at each stage

Every stage-1 outcome must have the SAME number of stage-2 branches.

Outcomes at tips

Read complete paths. Outcomes go at the END of branches, not the middle.

Count the leaves

The number of leaf nodes = total outcomes. Check matches multiplication.

The Multiplication Principle

+5 XP

When a multi-stage experiment has independent stages, the total number of outcomes = product of the options at each stage. This works for any number of stages and is much faster than drawing the full tree for large problems.

A restaurant has 3 starters, 4 mains, and 2 desserts. Total meals = 3 × 4 × 2 = 24 meals. Drawing the full tree would have 24 leaf nodes — multiplication is much faster. The key is that each stage is independent (your starter choice doesn't limit your main choice).

Total = 3 × 4 × 2 = 24

Independent stages

The multiplication principle assumes each stage doesn't change the options at other stages.

Extend to 3+ stages

Keep multiplying: stage 1 × stage 2 × stage 3 × ...

Much faster than listing

For 5 shirts × 3 pants × 4 hats = 60. Drawing the tree would take forever!

Watch Me Solve It · 3 examples

Watch Me Solve It · Tree diagram for two coin flips

+15 XP per step

PROBLEM

Draw a tree diagram for flipping a fair coin twice. List all outcomes and find P(at least one tail).

1

Draw Stage 1 branches

From Start: H branch and T branch

Two outcomes at stage 1: heads or tails.
2

Draw Stage 2 branches and list outcomes

From H: HH, HT From T: TH, TT Total: 4 outcomes
3

Find P(at least one tail)

At least one tail: {HT, TH, TT} = 3 outcomes. P = 3/4

Or use complement: P(no tails) = P(HH) = 1/4. So P(at least one tail) = 1 − 1/4 = 3/4.

AnswerSample space: {HH, HT, TH, TT}. P(at least one tail) = 3/4.

Watch Me Solve It · Die + coin outcomes

+15 XP per step

PROBLEM

A die (1–6) is rolled and a coin is tossed. How many outcomes are in the sample space? List them using the multiplication principle.

1

Identify options at each stage

Stage 1 (die): 6 options Stage 2 (coin): 2 options

These stages are independent — the die result doesn't affect the coin.
2

Apply multiplication principle

Total = 6 × 2 = 12 outcomes
3

Verify by listing a few

{1H, 1T, 2H, 2T, 3H, 3T, 4H, 4T, 5H, 5T, 6H, 6T} = 12 outcomes ✓

12 outcomes listed, matches the multiplication.

Answer12 outcomes: {1H, 1T, 2H, ..., 6T}. P(any specific outcome) = 1/12.

Watch Me Solve It · Outfit combinations

+15 XP per step

PROBLEM

A wardrobe has 3 shirts (red, blue, white) and 4 pairs of pants (black, grey, navy, beige). How many different outfits are possible?

1

Identify the two stages

Stage 1: shirts = 3 options Stage 2: pants = 4 options

Choosing a shirt does not limit which pants you can choose.
2

Apply multiplication principle

Total = 3 × 4 = 12 outfits
3

List some to verify

Red-black, Red-grey, Red-navy, Red-beige (4 outfits for red shirt alone), × 3 shirts = 12

Each shirt pairs with 4 pants. 3 shirts × 4 = 12. Confirmed!

Answer3 × 4 = 12 possible outfits.

Common Pitfalls

heads-up

Listing HT and TH as one outcome

Students often see "one head and one tail" as a single outcome. But the order matters: H then T is a different sequence from T then H. In probability, HT and TH are separate outcomes.

Fix: When drawing the tree, every path from left to right is distinct. Never combine symmetric paths.

Not completing all branches at each stage

A common mistake is drawing branches only for some stage-1 outcomes. Every outcome at stage 1 must have a full set of stage-2 branches — every single one.

Fix: Count the branches at stage 1. Each should have exactly the same number of stage-2 branches. If they differ, something is missing.

Adding instead of multiplying for total outcomes

2 shirt choices + 3 pants = 5 is wrong. You don't add the options at each stage — you multiply them. Correct: 2 × 3 = 6 outfits.

Fix: Multiplication principle: total = stage1 × stage2. Think: each shirt pairs with each pair of pants separately.

Copy Into Your Books

Sample Space

All possible outcomes listed
Must be complete and systematic
Order matters in multi-stage

Tree Diagrams

Branch for each option at each stage
Outcomes listed at leaf tips
Read complete paths left → right

Multiplication Principle

Total = stage1 × stage2 × ...
Stages must be independent
Faster than listing for large problems

Grid Method

Rows = stage 1 outcomes
Columns = stage 2 outcomes
Each cell = one combined outcome

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Sample Space and Tree Diagrams

4 problems

Four drill problems. Work each, then reveal the answer.

1 Three coins are flipped at once. How many outcomes are in the sample space?

2 × 2 × 2 = 8. Sample space: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}.8 outcomes
2 Draw a tree diagram for choosing from {A, B} then {1, 2, 3}. List all 6 outcomes.

From A: A1, A2, A3. From B: B1, B2, B3. Total = 6 = 2 × 3.{A1, A2, A3, B1, B2, B3}
3 What is P(getting A3) from the {A,B} × {1,2,3} experiment?

1 favourable outcome (A3) out of 6 total equally likely outcomes.P(A3) = 1/6
4 A school canteen has 3 choices of main and 2 choices of dessert. How many different lunch combinations are possible?

3 mains × 2 desserts = 6 combinations.3 × 2 = 6 lunch combinations

Complete in your workbook.

Quick Check · 5 questions

How many outcomes are in the sample space for flipping a coin twice?

+10 XP

4 different shirts and 3 different pants. How many outfits?

+10 XP

In a tree diagram, where are the outcomes written?

+10 XP

A restaurant has 3 entrees, 2 mains, and 4 desserts. How many 3-course meal combinations?

+10 XP

Three fair coins are flipped. How many outcomes in the sample space?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Medium 3 MARKS

Q6. A spinner with outcomes {Red, Blue, Green} is spun and a fair coin is tossed. (a) Draw a tree diagram. (b) List all outcomes. (c) Find P(Blue, Tails).

Draw the tree diagram in your workbook.

Understand Easy 2 MARKS

Q7. A PIN code uses 2 digits, each chosen from {1, 2, 3, 4, 5} and digits can repeat. How many PIN codes are possible? Show your reasoning.

Answer in your workbook.

Reason Hard 4 MARKS

Q8. A coin is flipped and a die (1–6) is rolled. (a) How many outcomes? (b) List all outcomes where the result is (heads, even number). (c) What is the probability of this event? (d) Why can't you simply add the number of coins and die faces?

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — 4 outcomes: HH, HT, TH, TT. HT ≠ TH.

2. C — 4 × 3 = 12 outfits.

3. D — At the tips of the final branches.

4. A — 3 × 2 × 4 = 24.

5. B — 2 × 2 × 2 = 8 outcomes.

Show Your Working Model Answers

Q6 (3 marks): (a) Tree: from Start, three branches R/B/G. From each, two branches H/T. (b) 6 outcomes: {RH, RT, BH, BT, GH, GT} [1]. Total = 3 × 2 = 6 [1]. (c) P(Blue, Tails) = P(BT) = 1/6 [1].

Q7 (2 marks): Stage 1: 5 options. Stage 2: 5 options (can repeat) [1]. Total = 5 × 5 = 25 PIN codes [1].

Q8 (4 marks): (a) 2 × 6 = 12 outcomes [1]. (b) {H2, H4, H6} — 3 outcomes [1]. (c) P = 3/12 = 1/4 [1]. (d) Adding gives 2+6=8 which is wrong. The multiplication principle applies because each coin outcome pairs with each die outcome independently. Adding only works if outcomes are mutually exclusive, which they're not here [1].

Stretch Challenge · +25 XP, +10 coins

The Locker Code

A school locker has a 3-digit code where each digit is from 1–4. (a) How many codes are possible if digits can repeat? (b) How many if no digit can repeat? (c) What fraction of all codes contain at least one "4"? (d) A student guesses randomly — what is P(guessing correctly)?

Reveal solution

(a) 4 × 4 × 4 = 64 codes. (b) 4 × 3 × 2 = 24 codes (no repeats). (c) Codes with NO 4: 3 × 3 × 3 = 27. Codes with at least one 4 = 64 − 27 = 37. Fraction = 37/64. (d) P(correct guess) = 1/64 ≈ 0.016.

Quick Review

Sample space

ALL possible outcomes — must be complete

HT ≠ TH

Order matters — they're different outcomes

Tree diagram

Branch at each stage, outcomes at tips

Multiplication

Total = stage1 × stage2 × ...

Grid method

Rows × columns = all combinations

Verify

Count leaves = multiplication result

Interactive: Tree Diagram Builder

Build your own tree diagrams by selecting options at each stage. See how the sample space grows with each additional stage.

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