11-15-2025, 04:33 PM
Chapter 8 — Tree Diagrams
Tree diagrams are one of the MOST powerful tools in probability.
They help you:
• organise information
• visualise steps
• see dependent and independent events
• calculate combined probabilities with ease
Students who learn tree diagrams properly score MUCH higher on probability questions.
---
8.1 What a Tree Diagram Actually Is
A tree diagram is simply a map of:
• every event
• every possible outcome
• their probabilities
At each “branch,” you multiply probabilities.
At the end of each “path,” you add probabilities if combining outcomes.
---
8.2 Key Rules
1. Probabilities on branches of the same step must add to 1.
2. Multiply along the branches (across).
3. Add vertically for final outcomes if needed.
4. If events depend on earlier outcomes → adjust the totals for the second branch.
---
8.3 Simple Example — Independent Events
A coin is flipped twice.
Tree diagram branches:
Flip 1:
• H = 1/2
• T = 1/2
Flip 2 (same probabilities):
• H = 1/2
• T = 1/2
Total paths:
• H → H
• H → T
• T → H
• T → T
Probability of each path:
(1/2 × 1/2) = 1/4
---
8.4 Example — Dependent Events (Without Replacement)
A bag has:
• 3 red
• 2 blue
Pick one, do NOT replace.
Then pick again.
Step 1
P(red) = 3/5
P(blue) = 2/5
Step 2 depends on Step 1:
If first was red → remaining: 2 red, 2 blue
P(red second | red first) = 2/4 = 1/2
P(blue second | red first) = 2/4 = 1/2
If first was blue → remaining: 3 red, 1 blue
P(red second | blue first) = 3/4
P(blue second | blue first) = 1/4
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8.5 Example — Probability of Two Reds
Using the above tree:
P(two red)
= (3/5 × 2/4)
= 6/20
= 3/10
---
8.6 Combined Events
If a question asks for:
"At least one red"
You add the paths that match the requirement.
For example (using previous numbers):
Paths with ≥1 red:
• R → R
• R → B
• B → R
Find each path probability and add them.
---
8.7 Tree Diagrams With Conditional Information
These questions combine the last chapter with this one.
Example:
A survey shows:
• 40% like tea
• 60% do not
Of those who like tea:
• 30% also like coffee
Of those who do not like tea:
• 10% like coffee
Tree diagram:
Step 1: Tea or Not Tea
Step 2: Coffee or No Coffee (probabilities differ)
Example calculation:
P(likes tea AND coffee)
= 0.40 × 0.30
= 0.12
---
8.8 Exam Example (Classic)
A bag contains:
• 4 chocolate sweets
• 6 fruit sweets
Two are taken at random without replacement.
Find the probability that:
(a) both are chocolate
(b) exactly one is chocolate
Solution Outline:
Step 1
P© = 4/10
P(F) = 6/10
Step 2 (dependent)
If C first → remaining: 3C, 6F
P© = 3/9
P(F) = 6/9
If F first → remaining: 4C, 5F
P© = 4/9
P(F) = 5/9
Final calculations:
(a) Both chocolate:
(4/10 × 3/9) = 12/90 = 2/15
(b) Exactly one chocolate:
Paths: C→F OR F→C
(4/10 × 6/9) + (6/10 × 4/9)
= 24/90 + 24/90
= 48/90
= 8/15
---
8.9 Mixed Tree & Ratio Problems
GCSE examiners LOVE this twist.
Example:
A class has:
• 12 boys
• 18 girls
Ratio wearing glasses:
• Boys: 1 in 4
• Girls: 1 in 6
Tree diagram:
Step 1: Boy / Girl
Step 2: glasses / no glasses
Example question:
Find P(student is a girl who wears glasses)
Girls = 18/30
Girls with glasses = 1/6
P = 18/30 × 1/6
= 18/180
= 1/10
---
8.10 Summary of How to Use Tree Diagrams
Tree diagrams are the SINGLE BEST TOOL for:
• dependent probability
• conditional probability
• multi-step events
• exam questions involving ratios
• “at least one” problems
• no-replacement problems
Remember the rules:
1. Multiply across
2. Add down
3. Change probabilities if totals change
4. Always check if the question is dependent
---
8.11 Your Turn — Practice Questions
1. A bag has 3 red, 2 blue.
Two are chosen without replacement.
Find P(both red).
2. A spinner lands on green 40% of the time.
If it lands on green: 25% chance of bonus spin.
If not green: 5% chance of bonus spin.
Find P(get bonus spin).
3. A box has 5 working pens and 3 faulty pens.
Two are selected without replacement.
Find P(exactly one faulty).
4. A class has 10 boys and 20 girls.
5 boys like football.
8 girls like football.
Find P(student is girl AND likes football).
5. A bag has 2 blue, 3 red, 1 yellow.
Two sweets are chosen without replacement.
Find P(at least one red).
---
Chapter Summary
• Tree diagrams VISUALLY map probability
• Use multiplication for combining steps
• Use addition for combining outcomes
• Dependent events change totals in step 2
• Most exam problems can be solved instantly using a well-drawn tree
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive
Tree diagrams are one of the MOST powerful tools in probability.
They help you:
• organise information
• visualise steps
• see dependent and independent events
• calculate combined probabilities with ease
Students who learn tree diagrams properly score MUCH higher on probability questions.
---
8.1 What a Tree Diagram Actually Is
A tree diagram is simply a map of:
• every event
• every possible outcome
• their probabilities
At each “branch,” you multiply probabilities.
At the end of each “path,” you add probabilities if combining outcomes.
---
8.2 Key Rules
1. Probabilities on branches of the same step must add to 1.
2. Multiply along the branches (across).
3. Add vertically for final outcomes if needed.
4. If events depend on earlier outcomes → adjust the totals for the second branch.
---
8.3 Simple Example — Independent Events
A coin is flipped twice.
Tree diagram branches:
Flip 1:
• H = 1/2
• T = 1/2
Flip 2 (same probabilities):
• H = 1/2
• T = 1/2
Total paths:
• H → H
• H → T
• T → H
• T → T
Probability of each path:
(1/2 × 1/2) = 1/4
---
8.4 Example — Dependent Events (Without Replacement)
A bag has:
• 3 red
• 2 blue
Pick one, do NOT replace.
Then pick again.
Step 1
P(red) = 3/5
P(blue) = 2/5
Step 2 depends on Step 1:
If first was red → remaining: 2 red, 2 blue
P(red second | red first) = 2/4 = 1/2
P(blue second | red first) = 2/4 = 1/2
If first was blue → remaining: 3 red, 1 blue
P(red second | blue first) = 3/4
P(blue second | blue first) = 1/4
---
8.5 Example — Probability of Two Reds
Using the above tree:
P(two red)
= (3/5 × 2/4)
= 6/20
= 3/10
---
8.6 Combined Events
If a question asks for:
"At least one red"
You add the paths that match the requirement.
For example (using previous numbers):
Paths with ≥1 red:
• R → R
• R → B
• B → R
Find each path probability and add them.
---
8.7 Tree Diagrams With Conditional Information
These questions combine the last chapter with this one.
Example:
A survey shows:
• 40% like tea
• 60% do not
Of those who like tea:
• 30% also like coffee
Of those who do not like tea:
• 10% like coffee
Tree diagram:
Step 1: Tea or Not Tea
Step 2: Coffee or No Coffee (probabilities differ)
Example calculation:
P(likes tea AND coffee)
= 0.40 × 0.30
= 0.12
---
8.8 Exam Example (Classic)
A bag contains:
• 4 chocolate sweets
• 6 fruit sweets
Two are taken at random without replacement.
Find the probability that:
(a) both are chocolate
(b) exactly one is chocolate
Solution Outline:
Step 1
P© = 4/10
P(F) = 6/10
Step 2 (dependent)
If C first → remaining: 3C, 6F
P© = 3/9
P(F) = 6/9
If F first → remaining: 4C, 5F
P© = 4/9
P(F) = 5/9
Final calculations:
(a) Both chocolate:
(4/10 × 3/9) = 12/90 = 2/15
(b) Exactly one chocolate:
Paths: C→F OR F→C
(4/10 × 6/9) + (6/10 × 4/9)
= 24/90 + 24/90
= 48/90
= 8/15
---
8.9 Mixed Tree & Ratio Problems
GCSE examiners LOVE this twist.
Example:
A class has:
• 12 boys
• 18 girls
Ratio wearing glasses:
• Boys: 1 in 4
• Girls: 1 in 6
Tree diagram:
Step 1: Boy / Girl
Step 2: glasses / no glasses
Example question:
Find P(student is a girl who wears glasses)
Girls = 18/30
Girls with glasses = 1/6
P = 18/30 × 1/6
= 18/180
= 1/10
---
8.10 Summary of How to Use Tree Diagrams
Tree diagrams are the SINGLE BEST TOOL for:
• dependent probability
• conditional probability
• multi-step events
• exam questions involving ratios
• “at least one” problems
• no-replacement problems
Remember the rules:
1. Multiply across
2. Add down
3. Change probabilities if totals change
4. Always check if the question is dependent
---
8.11 Your Turn — Practice Questions
1. A bag has 3 red, 2 blue.
Two are chosen without replacement.
Find P(both red).
2. A spinner lands on green 40% of the time.
If it lands on green: 25% chance of bonus spin.
If not green: 5% chance of bonus spin.
Find P(get bonus spin).
3. A box has 5 working pens and 3 faulty pens.
Two are selected without replacement.
Find P(exactly one faulty).
4. A class has 10 boys and 20 girls.
5 boys like football.
8 girls like football.
Find P(student is girl AND likes football).
5. A bag has 2 blue, 3 red, 1 yellow.
Two sweets are chosen without replacement.
Find P(at least one red).
---
Chapter Summary
• Tree diagrams VISUALLY map probability
• Use multiplication for combining steps
• Use addition for combining outcomes
• Dependent events change totals in step 2
• Most exam problems can be solved instantly using a well-drawn tree
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive