GCSE Maths • Foundation & Higher
GCSE Tree Diagrams Revision Guide
Organise every outcome, follow each route and calculate multi-stage probabilities accurately.
Foundation
Exam practice
Worked examples
What You'll Learn
Draw Complete Trees
Show every outcome at each stage and label every branch.
Use Branch Complements
Find a missing branch probability using 1−p.
Multiply Along Routes
Calculate the probability of a sequence of linked outcomes.
Add Successful Routes
Combine mutually exclusive ways in which the target event can occur.
Identify Independence
Use algebra to prove statements are always true.
Handle Conditional Probability
Update branch probabilities after events without replacement.
Key Rules and Formulas
Tree Diagrams Lessons
Conditional Probability Tree Diagrams
Update later probabilities for dependent events, especially selection without replacement.
Independent Events
Recognise unaffected events and multiply probabilities for repeated independent trials.
CORE SUBTOPICS
Tree Structure
Use nodes, labelled branches and endpoints to show the sample space.
Complementary Outcomes
Opposite outcomes on two branches total 1.
Outcome Order
Distinguish red-then-blue from blue-then-red.
Replacement
Return the selected item so later branch probabilities repeat.
No Replacement
Update both numerator and denominator after each selection.
Reverse Tree Problems
Use a stated final probability to form and solve an equation for a missing branch value.
Worked Example
Two Selections Without Replacement
QUESTION
A bag contains 7 red counters and 3 blue counters. Two counters are selected without replacement. Find the probability of selecting exactly one blue counter.
- Given information: Exactly one blue can occur in two mutually exclusive orders: blue then red, or red then blue.
- Method: Calculate each complete route by multiplication, then add the two route probabilities.
- Blue then red: 3/10 × 7/9 = 21/90 = 7/30.
- Red then blue: 7/10 × 3/9 = 21/90 = 7/30.
- Add the successful routes: 7/30+7/30=14/30.
- Simplify: 14/30=7/15.
- Check with the complement: 1−P(two red)−P(two blue)=1−(7/10×6/9)−(3/10×2/9)=1−7/15−1/15=7/15.
- Final answer: 7/15.
- Independent answer check: The direct route method and the complement method both give 7/15.
- One common mistake: After the first counter is removed, the second denominator is 9, not 10.
- Common GCSE Mistakes
- Incomplete Branches
- Branches Not Totalling One
- Adding Along a Route
- Multiplying Alternative Routes
- Ignoring Order
- Keeping Probabilities Unchanged
- Changing Probabilities with Replacement
- Assuming Every Pair Is Independent
- Missing Complement Routes
- Rounding Too Early
- Exam Tips
- Write probabilities directly on branches, not only at endpoints.
- Check each set of branches totals 1 before calculating.
- Highlight or trace every successful complete route.
- Write × signs along routes and + signs between alternative routes.
- Circle “with replacement” or “without replacement” in the question.
- For no replacement, update the denominator and the appropriate numerator.
- Use exact fractions and simplify the final result.
- For “at least one”, test whether the complement is quicker.
- Check that the final probability lies between 0 and 1.
Ready to Practice?
Printable Worksheet
Download and print practice questions with answers.
Interactive Quiz
Test your knowledge with instant feedback and hints.
Exam-Style Questions
Timed questions to build confidence for the real exam.
Watch the Tree Diagrams Tutorial
Clear, step-by-step explanation of expanding brackets, simplifying complex expressions, and solving for x. Perfect for visual learners who need a breakdown of the rules in action.
- 12 minute comprehensive guide
- High-definition whiteboard walk-through
Continue Your GCSE Maths Revision
Frequently Asked Questions
A branching diagram showing the possible outcomes and probabilities of successive events.
All branches leaving the same point must add to 1.
Multiply along one complete route to find the probability of all events on that route.
Add separate mutually exclusive routes that each satisfy the question.
