Tree Diagrams
Practise Tree Diagrams for GCSE Statistics with this free worksheet and full mark scheme — Foundation and Higher exam-style questions with worked answers for AQA and Edexcel. Tree diagrams show the outcomes and probabilities of a sequence of events.
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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA and Edexcel specifications. Every worksheet comes with a full mark scheme.
Topic overview
A tree diagram shows the possible outcomes of two or more successive events, with probabilities written on each branch.
Each set of branches from a point must have probabilities summing to 1, which provides a check at every stage. To find the probability of a particular path, multiply along its branches. To find the probability of several paths, add the results.
For events without replacement, the probabilities on the second set of branches differ depending on which first branch was taken. This is where tree diagrams earn their value: they force you to write each adjusted probability down rather than trying to hold them in your head.
Revision notes
Building the diagram
Draw one set of branches for each event, with the probability written on each branch.
Each set of branches from a single point must have probabilities summing to 1. Checking this at every stage catches errors early.
Reading the diagram
Multiply along the branches of a path to find its probability.
Add the probabilities of different paths when more than one satisfies the condition. The probabilities of all complete paths sum to 1, which is a final check.
Without replacement
The second set of branches has different probabilities depending on the first branch taken.
After taking a red, fewer reds remain; after taking a blue, all the reds remain. Writing each adjusted probability on its own branch prevents the most common error.
Key points
- Tree diagrams show successive events.
- Probabilities are written on each branch.
- Branches from a point sum to 1.
- Multiply along a path.
- Add the probabilities of different paths.
- Second-stage probabilities differ without replacement.
Worked examples
Example 1
A tree diagram has first branches with probability 0.4. Work out the probability on the other first branch. [2 marks]
Working
Example 2
On a tree diagram, one path has probabilities 0.3 and 0.5. Work out the probability of that path. [2 marks]
Working
Example 3
Explain why the second set of branches differs between paths when there is no replacement. [2 marks]
Working
Common mistakes
Adding along a path instead of multiplying.
Multiply along branches; add between paths.
Using the same second-stage probabilities on every path.
Without replacement they differ by path.
Not checking branches sum to 1.
This catches errors before they propagate.
Missing a path that satisfies the condition.
List all qualifying paths before adding.
Exam tips
- Check each set of branches sums to 1.
- Multiply along, add between.
- Write adjusted probabilities on each second-stage branch.
- List every qualifying path before adding.
Key terms
- Tree diagram
- A diagram showing outcomes of successive events.
- Branch
- One possible outcome, labelled with its probability.
- Path
- A route through the diagram giving a combined outcome.
- Successive events
- Events happening one after another.
Related topics
Written and reviewed against the current AQA and Edexcel specifications. Spotted an error? Let us know.