GCSE Maths → Probability → Tree Diagrams
Tree Diagrams GCSE Maths
Tree diagrams are used in GCSE probability to show possible outcomes in stages. They are especially useful when a question involves two events happening one after another.
Before learning tree diagrams, it helps to understand probability basics and fractions.
Video explanation
A short Worthing Maths Tutor video explanation for probability tree diagrams GCSE maths can be embedded here later to improve student engagement and time on page.
What is a tree diagram?
A tree diagram shows each possible outcome as a branch. Each branch is labelled with a probability.
Multiplying along branches
When finding the probability of one route through a tree diagram, multiply along the branches.
Example 1: Two coin tosses
A fair coin is tossed twice. Find the probability of getting two heads.
Adding different successful routes
If more than one route gives the required outcome, find each route first, then add them.
Example 2: One head and one tail
A fair coin is tossed twice. Find the probability of getting one head and one tail in any order.
The successful routes are HT and TH.
Tree diagrams without replacement
Some questions involve choosing objects without replacement. This means the probabilities change after the first choice.
Example 3: Counters without replacement
A bag contains 3 red counters and 2 blue counters. Two counters are chosen without replacement. Find the probability of choosing two red counters.
First red:
After one red is chosen, there are 2 red counters left out of 4 counters.
Second red:
Multiply along the route:
Tree diagrams with replacement
With replacement means the item is put back, so the probabilities stay the same on the second choice.
Example 4: Counters with replacement
A bag contains 3 red counters and 2 blue counters. A counter is chosen, replaced, then another counter is chosen. Find the probability of two red counters.
Common mistakes in tree diagrams
- Adding along branches instead of multiplying.
- Forgetting to add different successful routes.
- Using the same probabilities in without replacement questions.
- Forgetting that branch probabilities from the same point add to 1.
- Not simplifying the final fraction.
Practice questions
- A fair coin is tossed twice. Find the probability of two tails.
- A fair coin is tossed twice. Find the probability of exactly one head.
- A bag contains 4 red and 6 blue counters. One counter is chosen, replaced, then another is chosen. Find P(two red).
- A bag contains 4 red and 6 blue counters. Two counters are chosen without replacement. Find P(two red).
- A spinner has probability 0.3 of landing on red. It is spun twice. Find P(red both times).
Answers
- 1/4
- 1/2
- 4/10 × 4/10 = 4/25
- 4/10 × 3/9 = 2/15
- 0.09
Tree diagrams FAQ
What is a probability tree diagram?
A probability tree diagram shows possible outcomes in stages and helps calculate probabilities for combined events.
When do you multiply probabilities on a tree diagram?
Multiply probabilities along branches when finding the probability of events happening together.
When do you add probabilities on a tree diagram?
Add probabilities from different routes when more than one route gives the required outcome.
Are tree diagrams on GCSE maths?
Yes. Tree diagrams are common on GCSE maths papers, especially for combined events and conditional probability.
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