GCSE Maths Practice: tree-diagrams

Higher Tree Diagrams: “Exactly One” Events

Higher-tier tree diagram questions often ask for probabilities such as exactly one, at least one, or no more than one. These questions are harder because you usually need to consider multiple paths on the tree diagram and then add the probabilities of those paths. This is different from simpler questions where you multiply along just one branch.

Why a Tree Diagram Helps

When events happen in sequence (first draw, second draw, third draw), a tree diagram shows how probabilities change after each step. With no replacement, the totals drop by 1 each time and the numerator changes depending on what has already been taken. The diagram keeps this organised and reduces the chance of mixing up denominators.

Using “Green” and “Not Green”

In some questions, you can simplify the tree by grouping outcomes. For example, if you only care whether a marble is green or not, you can use two branches at each stage: Green and Not Green. This still produces a valid tree diagram because all non-green outcomes (red and blue) can be treated as one combined group when only the green count matters. This keeps the tree clear even with three stages.

How “Exactly One” Works

“Exactly one green in three draws” means:

• One draw is Green
• The other two draws are Not Green

So you list all the valid sequences and add them. For three draws, the sequences are typically patterns like: GNN, NGN, and NNG. Each sequence is a different path on the tree, and each path probability is found by multiplying along that route.

Worked Example (Different Question)

A bag contains 1 gold counter and 4 silver counters. Three counters are drawn without replacement. Find the probability of getting exactly one gold counter.

• Possible sequences: GSS, SGS, SSG
• Multiply each path probability
• Add the three results

This is the same method: identify the correct paths, multiply along them, then add.

Common Mistakes

• Forgetting to include all valid sequences (missing a path)
• Adding probabilities too early instead of multiplying along each path first
• Not updating denominators correctly (10 → 9 → 8 in three draws)
• Mixing up “exactly one” with “at least one”

Exam Tip

For “exactly one” questions, always write the sequences first. Then tick them off as you calculate each path. This prevents missing a case and makes your final answer much more reliable.