Tree Diagrams
A tree diagram is a branching diagram used to show the probabilities of two or more events happening one after another, with each branch labelled with a probability and each path along the branches representing one possible combined outcome. In GCSE Statistics, tree diagrams help calculate the probability of combined events by multiplying along branches and adding between separate paths.
Before you start
Make sure you're comfortable with these topics first:
Method
- Draw a first set of branches for the outcomes of the first event, and label each branch with its probability.
- From the end of each first branch, draw a second set of branches for the outcomes of the second event, again labelling each with its probability.
- Check that the probabilities on any set of branches from the same point add up to 1.
- To find the probability of one particular path, multiply the probabilities along that path.
- To find the probability of several different paths that all satisfy the question, calculate each path's probability separately then add them together.
- If the events are without replacement, remember the probabilities can change on later branches, since the total number of items available has decreased.
Worked example
A bag contains 5 red counters and 3 blue counters. A counter is taken at random and not replaced, then a second counter is taken at random. Find the probability that both counters are red.
- There are 8 counters in total, so P(first counter red) = 5/8.
- Since the first counter is not replaced, only 7 counters remain, of which 4 are red, so P(second counter red, given first was red) = 4/7.
- Multiply along the branches: P(both red) = 5/8 x 4/7.
- Multiply the fractions: (5 x 4)/(8 x 7) = 20/56.
- Simplify the fraction: 20/56 = 5/14.
- Final answer: P(both red) = 5/14.
Practice questions
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Q1On a tree diagram, what do you do to find the probability of a single path through the branches?Show answer
Answer: Multiply the probabilities along that path.
Q2Three branches from a single point are labelled 0.4, 0.35 and x. Find x.Show answer
Answer: x = 0.25 (probabilities from one point add up to 1: 1 - 0.4 - 0.35 = 0.25).
Q3A coin is flipped twice. Find the probability of getting two heads.Show answer
Answer: 1/4 (P(H) x P(H) = 1/2 x 1/2 = 1/4).
Q4A spinner has P(win) = 0.3. It is spun twice, independently. Find the probability of winning both times.Show answer
Answer: 0.09 (0.3 x 0.3 = 0.09).
Q5A bag has 4 green and 6 yellow balls. One ball is drawn, replaced, then a second is drawn. Find P(both yellow).Show answer
Answer: 9/25 (6/10 x 6/10 = 36/100 = 9/25).
Q6A box has 3 faulty bulbs out of 10. Two bulbs are chosen without replacement. Find the probability at least one is faulty.Show answer
Answer: 8/15 (P(no faulty) = 7/10 x 6/9 = 42/90 = 7/15; P(at least one faulty) = 1 - 7/15 = 8/15).
Exam-style questions
Written in the style of a GCSE Statistics exam paper, with a full mark scheme.
A weather forecaster says the probability it rains on Saturday is 0.4. If it rains on Saturday, the probability it also rains on Sunday is 0.6; if it does not rain on Saturday, the probability it rains on Sunday is 0.2. Find the probability it rains on both days.
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Using the same forecaster's figures (P(rain Saturday) = 0.4; P(rain Sunday given rain Saturday) = 0.6; P(rain Sunday given no rain Saturday) = 0.2), find the probability that it rains on exactly one of the two days.
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Explain why, when two counters are picked from a bag without replacement, the probabilities on the second set of branches of a tree diagram are different depending on which branch was followed first.
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See real GCSE Statistics past-paper questions, with official mark schemes →
Free printable worksheet
Want more practice on paper? Download the tree diagrams worksheet pack - 19 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 21 of GCSE Statistics Foundation Workbook 2 and chapter 27 of GCSE Statistics Higher Workbook 2, the whole course as one free printable PDF.
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