A coin is flipped twice. What is the probability of getting HH?
Multiply probabilities along the branch.
1/2 × 1/2 = 1/4.
Tree Diagrams
Tree diagrams are used to represent possible outcomes in multi-step events. They help organise information clearly and calculate probabilities accurately, building on listing outcomes and supporting ideas in conditional probability.
Overview
A tree diagram shows all the possible outcomes of a multi-step probability experiment.
Each branch shows one possible path, and the probabilities are written on the branches.
Along a branch, multiply probabilities
Tree diagrams are especially useful for questions with two or more stages, such as tossing two coins, choosing counters, or repeated events.
What you should understand after this topic
- Draw and read a tree diagram
- Label branches with probabilities
- Multiply probabilities along a path
- Add probabilities of different successful outcomes
- Work with with-replacement and without-replacement questions
Key Definitions
Tree Diagram
A branching diagram that shows all possible outcomes in order.
Branch
A line showing one possible option at a stage.
Outcome
A final result at the end of a path, such as HH or RB.
Path
A route through the tree from start to end.
With Replacement
The item is put back, so probabilities stay the same for the next step.
Without Replacement
The item is not put back, so probabilities can change for the next step.
Key Rules
Multiply along branches
To find the probability of one path, multiply the branch probabilities.
Add separate successful paths
If more than one outcome works, add those path probabilities.
Branch totals must equal 1
At each split, the probabilities should add up to 1.
Check replacement carefully
With replacement = same probabilities. Without replacement = changed probabilities.
Quick Reminder
Along
<span class='tree-tag'>Multiply</span>
Across outcomes
<span class='tree-tag'>Add</span>
With replacement
Probabilities stay the same.
Without replacement
Probabilities may change.
How to Solve
Step 1: Understand tree diagrams
A tree diagram shows all possible outcomes for events that happen in stages.
Each full path gives one complete outcome.
Exam tip: Work from left to right.
This is a visual way of organising outcomes from listing outcomes.
Step 2: Draw and label branches
- Draw branches for the first event.
- From each branch, draw branches for the second event.
- Label each outcome clearly.
- Write probabilities on every branch.
Step 3: Multiply along a path
To find the probability of one complete outcome, multiply along the branches.
\( P(HH) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \)
Step 4: Add successful paths
If more than one outcome works, add the path probabilities.
\( P(HT \text{ or } TH) = \frac{1}{4} + \frac{1}{4} = \frac{1}{2} \)
Exam thinking: Multiply along branches, add different routes.
Step 5: With replacement
If the item is put back, the probabilities stay the same.
Example: picking a counter, replacing it, then picking again.
The total number of counters stays the same.
Step 6: Without replacement
If the item is not put back, the probabilities change.
Example: after one counter is removed, fewer counters are left.
Exam tip: Update both the numerator and denominator.
This idea is important for conditional probability.
Step 7: Check your tree diagram
Labels
Every branch should be labelled.
Probabilities
Each branch needs a probability.
Branch totals
Probabilities at each split should add to 1.
Final outcomes
Each route should represent one outcome.
Step 8: Exam method summary
See listing outcomes for simpler probability outcomes.
- Draw the branches for each stage.
- Write probabilities on every branch.
- Multiply along each route.
- Add routes that match the event.
- Check whether the question is with or without replacement.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics, focusing on completing and using probability tree diagrams.
Edexcel
A biased coin is flipped twice. The probability of getting heads on each flip is 0.6.
Complete a probability tree diagram for this information.
Edexcel
A biased coin is flipped twice. The probability of getting heads on each flip is 0.6.
Find the probability of getting two heads.
Edexcel
A biased coin is flipped twice. The probability of getting heads on each flip is 0.6.
Find the probability of getting exactly one head.
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on tree diagrams for events without replacement.
AQA
A bag contains 5 red counters and 3 blue counters. A counter is taken at random and not replaced. A second counter is then taken at random.
Complete a probability tree diagram for this information.
AQA
A bag contains 5 red counters and 3 blue counters. A counter is taken at random and not replaced. A second counter is then taken at random.
Find the probability that both counters are red.
AQA
A bag contains 5 red counters and 3 blue counters. A counter is taken at random and not replaced. A second counter is then taken at random.
Find the probability that the two counters are different colours.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising multiplication along branches and addition of different successful outcomes.
OCR
There are 4 green counters and 6 yellow counters in a bag. Two counters are taken at random without replacement.
Find the probability that both counters are yellow.
OCR
There are 4 green counters and 6 yellow counters in a bag. Two counters are taken at random without replacement.
Find the probability that one counter is green and one counter is yellow.
OCR
There are 4 green counters and 6 yellow counters in a bag. Two counters are taken at random without replacement.
Explain why the probabilities on the second set of branches are different from the first set of branches.
Exam Checklist
Step 1
Draw every possible branch clearly.
Step 2
Write the correct probability on each branch.
Step 3
Multiply along a path for one full outcome.
Step 4
Add the successful path probabilities if needed.
Most common exam mistakes
Operation mistake
Adding along a path instead of multiplying.
Replacement mistake
Using the wrong second-stage probabilities.
Missing path mistake
Forgetting one of the successful outcomes.
Branch total mistake
Not checking that probabilities at each split add to 1.
Common Mistakes
These are common mistakes students make when working with tree diagrams in GCSE Maths.
Adding instead of multiplying along branches
Incorrect
A student adds probabilities along a path.
Correct
When following a path in a tree diagram, multiply the probabilities along the branches.
Forgetting to add final outcomes
Incorrect
A student finds only one path when multiple outcomes are required.
Correct
If more than one path leads to the desired outcome, add those probabilities together at the end.
Incorrect probabilities without replacement
Incorrect
A student keeps the same probabilities for the second stage.
Correct
In without-replacement questions, probabilities change after the first selection. Update totals and remaining outcomes.
Branch probabilities not summing to 1
Incorrect
A student writes probabilities that do not add up correctly.
Correct
At each split, the probabilities must add up to 1. Check each stage carefully.
Missing outcomes
Incorrect
A student does not include all possible paths.
Correct
Ensure all possible outcomes are represented in the tree diagram before calculating probabilities.
Try It Yourself
Practise solving probability problems using tree diagrams.
Foundation
Foundation Practice
Use tree diagrams to calculate probabilities of combined events.
A coin is flipped twice. Find the probability of HT.
Multiply probabilities.
1/2 × 1/2 = 1/4.
A bag has 1 red and 1 blue ball. One is picked, then replaced, then picked again. What is the probability of red then red?
Same probabilities each time.
1/2 × 1/2 = 1/4.
A fair die is rolled twice. What is the probability of getting a 3 both times?
Multiply probabilities.
1/6 × 1/6 = 1/36.
What do you do when moving along a branch in a tree diagram?
Branch = sequence.
Multiply along branches.
A coin is flipped twice. Find the probability of getting one head and one tail (HT or TH).
Add both outcomes.
HT = 1/4, TH = 1/4 → total = 1/2.
When finding total probability from multiple outcomes, what do you do?
Combine outcomes.
Add final probabilities.
A spinner has 2 equal sections: A and B. It is spun twice. Find the probability of AA.
Multiply probabilities.
1/2 × 1/2 = 1/4.
A student adds probabilities along a branch. What is wrong?
Branch = multiply.
Multiplication is used along branches.
A die is rolled twice. Find the probability of getting a 6 then a 6.
Multiply probabilities.
1/6 × 1/6 = 1/36.
Higher
Higher Practice
Solve tree diagram problems with and without replacement.
A bag has 3 red and 2 blue balls. One is picked, not replaced, then another is picked. What is the probability of red then red?
Probabilities change after first pick.
3/5 × 2/4 = 6/20 = 3/10.
A bag has 4 red and 1 blue ball. One is picked, not replaced. Find probability of red then blue.
Adjust totals after first pick.
4/5 × 1/4 = 4/20 = 1/5.
What changes when there is no replacement?
Total reduces.
Removing items changes probabilities.
A bag has 2 red and 3 blue balls. One is picked, replaced, then picked again. Find probability of blue then blue.
Same probabilities each time.
3/5 × 3/5 = 9/25.
A bag has 5 balls: 2 red, 3 blue. What is P(red then blue, no replacement)?
Multiply adjusted probabilities.
2/5 × 3/4 = 6/20 = 3/10.
A bag has 3 red and 2 blue balls. Find probability of at least one red in two picks with replacement.
Use complement.
P(no red) = blue-blue = 2/5 × 2/5 = 4/25 → 1 − 4/25 = 21/25.
Why is complement useful in tree diagrams?
Easier than listing all cases.
Complement often simplifies problems.
A bag has 4 red and 6 blue balls. One is picked, not replaced. Find probability of blue then blue.
Adjust totals.
6/10 × 5/9 = 30/90 = 1/3.
A student uses the same probability after removing a ball. What is wrong?
Total changes.
Probabilities must be updated.
A bag has 2 red and 1 blue ball. One is picked, not replaced. Find probability of getting one red and one blue (any order).
Add both cases.
RB = 2/3 × 1/2 = 1/3, BR = 1/3 × 2/2 = 1/3 → total = 2/3.
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What are tree diagrams used for?
Showing sequences of events.
How do you calculate probability?
Multiply along branches.
Why are they useful?
They organise complex problems.