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 only half-remember.
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 the standard visual method for multi-step probability in Cambridge IGCSE Mathematics (0580/0607). Whenever a question involves two or more stages — tossing coins, drawing balls, or spinning twice — examiners reward a clear tree with probabilities on branches and path products for combined 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 stages as branches; multiply along a path for “and” (combined outcome).
• Add the probabilities of separate paths that satisfy the same event (e.g. “exactly one head”).
• Without replacement means second-branch probabilities must use updated totals.
• Branches from one point must sum to 1.

What are Tree Diagrams in Cambridge IGCSE Maths?

Tree Diagrams are diagrams that map successive events as branches, with probabilities written on each branch. In Cambridge IGCSE Mathematics you draw trees for two or three stages, find the probability of a specific outcome by multiplying along branches, and find the probability of combined events by adding relevant path totals. They are especially useful for “exactly one” and “at least one” problems.

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.

Idea	What it means	How the exam uses it
First branches	All outcomes of stage 1	”First ball drawn”
Second branches	Conditional on stage 1	”Without replacement” changes fractions
Path product	Multiply along one complete path	P(R then B) = P(R) × P(B after R)
Sum paths	Add paths for “or” outcomes	”Exactly one red” = two paths

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

The safest method works for coins, spinners and bags.

1. Draw stage 1 branches with all outcomes and probabilities summing to 1.
2. From each stage-1 outcome, draw stage 2 branches with correct conditional probabilities.
3. Label each path at the end (e.g. HH, HT, TH, TT).
4. Multiply along a path to get P(that specific outcome).
5. Add path probabilities that match the event asked (e.g. exactly one H).
6. Simplify fractions in the final answer.

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.

Independent vs dependent trees: which approach does the question want?

Students lose marks by reusing the same branch probabilities after a draw without replacement. Use the wording to decide.

Situation	What to do	Typical signal words
Independent stages	Second branches same as first logic	”replaced”, “coin tossed twice”
Without replacement	Update numerators and denominators	”without replacement”
Three stages	Extend tree to third set of branches	”three balls drawn”
Exactly k successes	Add all paths with k of the outcome	”exactly two heads”

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

Most lost marks come from incomplete trees or forgetting to add paths. These are the command words you will see.

Command word / phrase	What the question wants	Typical tree stem
Draw a tree diagram	Full diagram with probabilities on branches	”Draw a tree diagram to show the two draws.”
Find the probability	Calculate from tree (may not require redraw)	“Find the probability both balls are blue.”
Work out	Method required — tree or equivalent	”Work out the probability of exactly one red.”
Show that	Prove given probability — tree supports working	”Show that the probability is 1/3.”
Write down	State probability from prior tree	”Write down the probability the second is green.”

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 P(exactly one head).” Paths HT and TH each have probability (1/2) × (1/2) = 1/4. Sum: 1/4 + 1/4 = 1/2. Mark-scheme reward: tree drawn, both paths identified.
2. “A bag has 2 red and 3 blue balls. Two are drawn without replacement. Find P(both red).” Path R-R: (2/5) × (1/4) = 2/20 = 1/10. Reward: second branch 1/4 not 2/5.
3. “Spinner: P(red) = 0.3, P(blue) = 0.7. Spun twice. Find P(both blue).” 0.7 × 0.7 = 0.49. Reward: multiply independent branches.

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

Trees implement the same rules as Probability Applications but make errors visible. For frequency data, use Venn Diagrams And Tables instead. 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

• Adding along a path instead of multiplying.
• Second-branch probabilities not updated after without replacement.
• Branches from one node not summing to 1.
• Missing a path when finding “exactly one” (there are usually two).
• Drawing a tree but not using it to calculate — method marks need the products shown.

When you need more support

If tree diagram questions keep tripping you up — especially three-stage or without-replacement trees — 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? The structure is simple: multiply along paths, add for separate outcomes. Marks are lost when branch probabilities are wrong after without replacement, or when a path is missed.

Do I always need to draw a tree diagram? Draw one when the question asks, or when there are two or more stages and you risk losing track. For simple independent events, multiplication may be enough if working is clear.

How do I find P(exactly one head) from a tree? Identify every path with exactly one H (HT and TH), multiply along each path, then add the results.

How do I revise Tree Diagrams effectively? Practise coins, bags and spinners with and without replacement, check branches sum to 1, then take the Tree Diagrams quiz.

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