tree-diagrams – GCSE Maths Practice (Question 8 of 11)

Higher Tree Diagrams: Three Dependent Draws

This question is GCSE Higher because it uses a three-stage tree diagram. With each draw the total number of balls changes, and the number of each colour can change depending on what was drawn. This increases the chance of errors, so a tree diagram is the safest way to organise the work.

Why the Probabilities Change

The balls are drawn without replacement. That means:

The denominator decreases each draw (10, then 9, then 8).
The numerator only changes for the colour you actually removed.
Other colours stay the same until they are drawn.

How to Build the Tree Diagram

For Higher tier, you should show enough branches to reflect the full process. A clear approach is:

Stage 1: split into Red / Not Red.
From the Red branch, Stage 2: split into Blue / Not Blue.
From Red then Blue, Stage 3: split into Green / Not Green.

This keeps the diagram readable while still being a genuine three-level tree.

Multiplying Along a Path

Once the tree is labelled, the probability of a sequence is found by multiplying along the path. For example:

P(R then B then G) = P(R) × P(B|R) × P(G|R and B)

Worked Example (Different Numbers)

A bag contains 2 red, 4 blue and 3 green balls. Three balls are drawn without replacement. Find the probability of drawing Blue then Green then Red.

P(B) = 4/9
After B: P(G|B) = 3/8
After B and G: P(R|B,G) = 2/7
Multiply: 4/9 × 3/8 × 2/7

Common Mistakes

Forgetting the denominator changes each draw
Reducing the wrong numerator (only the drawn colour changes)
Stopping after two stages (Higher questions often need three)
Trying to simplify too early and making arithmetic slips

Exam Tip

Write the totals under each stage of the tree (10 → 9 → 8). This simple habit prevents most errors in Higher tree-diagram questions.

Study tip: For any “A then B then C” question without replacement, think: three branches, three fractions, multiply along one path.

GCSE Maths Practice: tree-diagrams