Who this is for: Cambridge IGCSE Mathematics (0580/0607) students who want Tree Diagrams — drawing branches, multiplying along paths and adding outcomes — to become a reliable source of marks instead of a method they skip under exam pressure.
What query it owns: how to understand and revise Tree Diagrams in Cambridge IGCSE Mathematics.
Why this is safe: this page owns the Tree Diagrams revision-guide angle, while Tutopiya’s Tree Diagrams subtopic page owns the learning resource and the free Tree Diagrams quiz owns the practice.

Tree Diagrams are one of the most useful tools in the Probability unit of Cambridge IGCSE Mathematics (0580/0607). Whenever a question involves two or more stages — tossing coins, drawing balls, or picking cards — examiners reward a clear tree with probabilities on every branch and correct use of multiply along paths, add across outcomes. This guide explains exactly what the subtopic covers, how to handle the question types that actually appear, and where to practise each skill.

Key takeaways

• A tree diagram shows every possible outcome of multi-stage experiments as branches.
• Multiply probabilities along a path (AND — all events on that path happen).
• Add probabilities of separate paths that satisfy the same condition (OR).
• Without replacement means the second-set branch probabilities must use the reduced total.

What are Tree Diagrams in Cambridge IGCSE Maths?

Tree Diagrams are visual representations of combined probability experiments where each branch shows an outcome and its probability. In Cambridge IGCSE Mathematics you draw trees for two (sometimes three) stages, label every branch, then multiply along paths and add across paths to find the probability of a combined event. Examiners often award method marks for a correct tree even when arithmetic slips.

You can read the full explanation, worked examples and notes on Tutopiya’s Tree Diagrams subtopic page before you attempt questions.

The core ideas you must master

These four ideas appear again and again. Learn what each one means and the exam phrasing that signals it.

Rule	What it means	How the exam uses it
Branch labels	Every outcome at each stage shown	”Draw a tree diagram to show…”
Multiply along path	P(path) = product of branch probabilities	”Find the probability of two reds”
Add across paths	P(event) = sum of matching path products	”Find the probability of exactly one red”
Probabilities sum to 1	At each node, branches from that point total 1	Check after drawing

How to draw and use a tree diagram — step by step

The safest method works for every two-stage probability question.

1. Draw the first set of branches from a single point — label every outcome and its probability.
2. From the end of each first branch, draw second-stage branches with updated probabilities if without replacement.
3. Write the probability on every branch — fractions in lowest terms.
4. Identify paths that match the event asked (e.g. both red, exactly one head).
5. Multiply along each path, then add the products for all matching paths.

Once you have worked through a few, test yourself with the free Tree Diagrams quiz — it tells you fast whether the method has actually stuck.

With replacement vs without replacement on trees

Students lose marks by using the same probabilities on second branches when balls are not replaced.

Setup	Second-branch probabilities	Typical signal words
With replacement	Same as first stage	”replaced”, “coin tossed again”
Without replacement	Denominator reduced by 1	”not replaced”, “two cards drawn”
Independent trials	Second branches identical	Separate coin tosses, dice rolls
”At least one”	Often easier via 1 − P(none) on tree	”at least one head”

Tree Diagrams in past-paper wording: command words that matter

Most lost marks come from incomplete trees or forgetting to add paths for “exactly one” questions.

Command word / phrase	What the question wants	Typical stem
Draw a tree diagram	Full tree with probabilities on branches	”Draw a tree diagram to show the two draws.”
Find the probability	Numeric answer from the tree	”Find the probability that both are red.”
Work out / Calculate	Same as find — method from tree	”Work out the probability of exactly one blue.”
Show all possible outcomes	Tree or systematic list	”Show all possible outcomes of two spins.”
Find the probability of at least one	Sum relevant paths or use complement	”Find the probability of at least one 6.”

Worked exam-style stems (how to answer the wording)

Practising the wording — not just the branches — is what method marks reward.

1. “A fair coin is tossed twice. Draw a tree diagram and find the probability of two heads.” Paths: HH = ½ × ½ = ¼. Mark-scheme reward: tree drawn with four end paths.
2. “A bag has 2 red and 3 blue balls. Two are drawn without replacement. Find the probability both are red.” RR = (2/5) × (1/4) = 2/20 = 1/10. Reward: second branch uses 4 in denominator.
3. “Find the probability of exactly one head in two tosses.” HT + TH = ¼ + ¼ = ½. Reward: two paths identified and added.

When you can recognise the wording instantly, work the full set on the Probability topical past paper questions and the Tree Diagrams quiz to lock the method in.

How Tree Diagrams connect to the rest of Probability

Tree Diagrams visualise the same rules taught in Probability Applications — if you prefer tables or formulae, trees are the diagram alternative. They also support problems in Venn Diagrams and Tables when data is presented differently. When you are ready to mix topics, the Cambridge IGCSE Maths resource hub lets you move straight from a weak subtopic into the next.

Common mistakes students make

• Missing branches — every outcome at each stage must appear.
• Using the same denominator on second branches when balls are not replaced.
• Forgetting to add paths for “exactly one” questions.
• Branch probabilities at a node not summing to 1.
• Drawing a tree but not using it — marks need path products shown.

When you need more support

If tree diagram questions keep tripping you up — especially without replacement — work through the Probability topical past paper questions and the Tree Diagrams quiz to pinpoint the exact gap, then get focused help from a Cambridge IGCSE Maths tutor to fix it quickly.

Frequently asked questions

Are Tree Diagrams hard in Cambridge IGCSE Maths? Drawing the tree is straightforward. Marks are lost when second-branch probabilities are wrong or paths are not added correctly.

When should I use a tree diagram? Use a tree for two- or three-stage probability with clear sequential outcomes — especially without replacement.

What does multiply along, add across mean? Multiply branch probabilities on one complete path (AND). Add products from different paths that both satisfy the event (OR).

How do I revise Tree Diagrams effectively? Read the subtopic notes, draw a full tree on every question, then take the Tree Diagrams quiz.

Ready to master Cambridge IGCSE Maths Tree Diagrams?

Start with the Tree Diagrams subtopic page, then book a free trial with a Cambridge IGCSE Maths specialist to turn Tree Diagrams into guaranteed marks.

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