Draw branches for each stage, multiply along a path, add across paths, and adjust the second-stage numbers when there is no replacement. It is on Foundation and Higher, aimed at grade 5. Free questions with answers are on this page.
A tree diagram splits a two-step chance experiment into branches. Each pair of branches from a point adds to 1. Along a route, multiply. Different routes that count as 'success', add. With replacement the second-level probabilities copy the first; without replacement they change (one fewer, one less of that colour).
Label every branch with a probability, even the 'obvious' 1−p. A 4-mark question is usually two multiply-along-the-route calculations then an add. Fractions stay as fractions; cancel when you multiply, not in a way that breaks the add-to-1 check on a pair of branches.
Each set of branches from one point adds to 1. Independent: same probabilities on every matching pair of branches. Without replacement: the second-branch denominators drop by 1. AND multiplies along a path; OR adds complete paths.
Objectives
Multiply two branch probabilities.
Find both-red without replacement, and a path with replacement.
Find the probability of not taking a branch.
Key points
Draw a two-stage tree with probabilities on every branch, each pair of branches summing to 1.
Multiply along a path to find the probability of a combined outcome.
Add the probabilities of different paths that satisfy the same event (e.g. 'exactly one').
Use 1−P(none) for 'at least one'.
Worked examples
Example 1
A tree-diagram path has branches 1/2 and 1/3. What is the probability of that path?
2 marks
Identify tree diagram rule: multiply probabilities along a path
Multiply fractions: (1/2) × (1/3) = 1/6
State the combined probability: 1/6
Answer: 1/6
Common slips
Slip: Alistair added 1/2 + 1/3 = 5/6 Probabilities along a sequence of events multiply.
Slip: Alistair added numerators and denominators Multiply numerator by numerator, denominator by denominator.
Slip: Alistair wrote 5/6 The probability of this path is 1/6.
Example 2
Two counters are taken without replacement from a bag of 5 counters (4 red). Find P(both red).
2 marks
Calculate first draw probability: 4/5
Calculate second draw without replacement: 3/4
Multiply: (4/5) × (3/4) = 12/20 = 3/5
Answer: 3/5
Common slips
Slip: Brenda used 4/4 Initially there are 4 red out of 5 total.
Slip: Brenda used 4/5 again Both red count and total count reduce by 1.
Slip: Brenda added 4/5 + 3/4 That is more than 1. Multiply along a path.
Example 3
A branch has probability 1/3. What is the probability of not taking that branch?
3 marks
Recall that probabilities from a node must sum to 1
Subtract branch probability from 1: 1 - 1/3 = 2/3
State the complement probability: 2/3
Answer: 2/3
Common slips
Slip: Clifford wrote 1/3 Not taking a branch is its complement.
Slip: Chloe divided 1 by 3 Complement rule: P(not A) = 1 - P(A).
Slip: Clifford wrote 0 1 - 1/3 = 2/3.
Questions and answers
GCSE Maths probability trees questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
A tree-diagram path has branches 1/2 and 1/4. What is the probability of that path?
Show answer
Answer: 1/8
Multiply along path: (1/2) × (1/4).
Result is 1/8.
M1 — Multiply along path: (1/2) × (1/4).
A1 — Result is 1/8.
Question 2 Easy2 marks
A tree-diagram path has branches 2/3 and 1/5. What is the probability of that path?
Show answer
Answer: 2/15
Multiply along path: (2/3) × (1/5).
Result is 2/15.
M1 — Multiply along path: (2/3) × (1/5).
A1 — Result is 2/15.
Question 3 Medium2 marks
Two counters are taken without replacement from 4 counters (3 red). Find P(both red).
Show answer
Answer: 1/2
First draw: 3/4. Second draw: 2/3.
Multiply: (3/4) × (2/3) = 6/12 = 1/2.
M1 — First draw: 3/4. Second draw: 2/3.
A1 — Multiply: (3/4) × (2/3) = 6/12 = 1/2.
Question 4 Medium2 marks
Two counters are taken without replacement from 6 counters (5 red). Find P(both red).
Show answer
Answer: 2/3
First draw: 5/6. Second draw: 4/5.
Multiply: (5/6) × (4/5) = 20/30 = 2/3.
M1 — First draw: 5/6. Second draw: 4/5.
A1 — Multiply: (5/6) × (4/5) = 20/30 = 2/3.
Question 5 Hard2 marks
With replacement, probability 1/3 then 1/3. What is the probability of both?
Show answer
Answer: 1/9
Independent draws: (1/3) × (1/3).
Result is 1/9.
M1 — Independent draws: (1/3) × (1/3).
A1 — Result is 1/9.
Question 6 Hard2 marks
With replacement, probability 1/4 then 1/2. What is the probability of both?
Show answer
Answer: 1/8
Independent draws: (1/4) × (1/2).
Result is 1/8.
M1 — Independent draws: (1/4) × (1/2).
A1 — Result is 1/8.
Question 7 Exam-style3 marks
Two counters are taken without replacement from 6 counters (4 red). Find P(both red).
Show answer
Answer: 2/5
First draw is 4/6.
Second draw without replacement is 3/5.
Multiply: (4/6) × (3/5) = 12/30 = 2/5.
M1 — Second branch 3/5 (one fewer red and one fewer counter)
M1 — Multiply branch probabilities: (4/6) × (3/5)
A1 — 2/5 or 12/30
Common mistakes
Most lost marks on this topic are one slip.
Adding branch probabilities instead of multiplying
Events happening in sequence multiply along tree paths
Keeping the same denominator on the second draw when sampling without replacement
Without replacement, the total count drops by 1 on subsequent draws
Forgetting to subtract from 1 for the complement
The probability of NOT taking a branch is 1 minus that branch probability
Spot the slip
Two branches 1/2 and 1/3. Ada adds the fractions. What is the probability of that path?
1/6
5/6
2/5
Show the slip
1/6 Multiply along a path: 1/2 × 1/3 = 1/6.
Exam tips
This topic can appear on a calculator or a non-calculator paper. Show enough working for a method mark.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Label every branch, even the ones you will not use; the completed tree is usually worth a mark by itself.
Write the multiplication for each path in full — a correct product line earns method even if the final addition slips.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.