Tree diagrams
Construct tree diagrams for two-stage experiments and use them to find probabilities with and without replacement.
Worked examples
Listing outcomes using a tree diagram
Straightforward
Problem
A bag contains one red (R) marble and one blue (B) marble. A marble is drawn at random, replaced, and then a second marble is drawn. Draw a tree diagram and list all outcomes in the sample space.
1
Draw the first stage of the tree.
From the start, draw two branches — one labelled R and one labelled B.
2
Draw the second stage of the tree.
Since the marble is replaced, the same two options (R and B) are available for the second draw. From each first-stage branch, draw two more branches labelled R and B.
3
List all outcomes by following each path from start to end.
The four paths give outcomes: RR, RB, BR, BB.
Answer
Sample space . There are 4 outcomes.
Finding probability from a tree diagram (with replacement)
Moderate
Problem
A bag contains 2 red (R) and 1 blue (B) marble. A marble is drawn, replaced, then drawn again. Using a tree diagram, find .
1
Write probabilities on the first-stage branches.
,
2
Write probabilities on the second-stage branches.
Because the marble is replaced, the probabilities are the same for the second draw. Write and from each first-stage branch.
3
Multiply along each path to find the probability of each outcome.
, , ,
4
Add the probabilities for all outcomes that include at least one blue marble.
Answer
Finding probability from a tree diagram (without replacement)
Challenging
Problem
A bag contains 3 red (R) and 2 blue (B) marbles. Two marbles are drawn without replacement. Using a tree diagram, find .
1
Write probabilities on the first-stage branches.
,
2
Write probabilities on the second-stage branches. These change because the first marble is not replaced.
If first draw was R: 2 red and 2 blue remain from 4 total, so and . If first draw was B: 3 red and 1 blue remain from 4 total, so and .
3
Find the probability for each mixed-colour path.
4
Add the two probabilities.
Answer
Practise
Q1·Straightforward
A fair coin is tossed twice. A tree diagram shows all possible outcomes. How many outcomes are in the sample space?
Explanation
The tree has two stages. At each stage two branches are drawn (H or T). The outcomes are HH, HT, TH and TT, giving outcomes in total.
Q2·Straightforward
Cards labelled 1, 2 and 3 are placed in a bag. A card is drawn at random, replaced, and then a second card is drawn. A tree diagram is used to list all outcomes. How many outcomes are in the sample space?
Explanation
There are 3 outcomes at each stage. The tree shows outcomes in total: (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3).
Q3·Straightforward
A fair coin is tossed and then a spinner with 3 equal sections labelled A, B and C is spun. A tree diagram shows all possible outcomes. How many outcomes are in the sample space?
Explanation
The coin gives 2 outcomes and the spinner gives 3. The tree shows outcomes: (H,A), (H,B), (H,C), (T,A), (T,B), (T,C).
Q4·Straightforward
A fair coin is tossed and a four-sided die numbered 1 to 4 is rolled. A tree diagram shows all possible outcomes. How many outcomes in the sample space contain a Head?
Explanation
The Head branch leads to four outcomes: (H,1), (H,2), (H,3) and (H,4). There are 4 outcomes that contain a Head.
Q5·Moderate
A bag contains 2 red and 1 blue marble. A marble is drawn at random, replaced, then a second marble is drawn. Using a tree diagram, find . Give your answer as a fraction in simplest form.
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Explanation
Since the marble is replaced, each draw has . The red–red path gives:
Q6·Moderate
A fair coin is tossed and a standard 6-sided die is rolled. Using a tree diagram, find . Give your answer as a fraction in simplest form.
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Explanation
The Heads branch has probability . The 3 branch from Heads has probability .
Q7·Moderate
A bag contains 1 red, 1 blue and 1 green marble. A marble is drawn, replaced, then drawn again. Using a tree diagram, find . Give your answer as a fraction in simplest form.
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Explanation
There are 9 equally likely outcomes. Same-colour paths RR, BB and GG each have probability .
Q8·Moderate
A fair 6-sided die is rolled twice. Using a tree diagram, find . Give your answer as a fraction in simplest form.
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Explanation
Numbers less than 3 are 1 and 2, so for each roll.
Q9·Challenging
A bag contains 3 red and 2 blue marbles. Two marbles are drawn without replacement. Using a tree diagram, find . Give your answer as a fraction in simplest form.
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Explanation
First draw: . After one red marble is removed, 4 remain with 2 red: .
Q10·Challenging
A bag contains 2 red and 2 blue counters. Two counters are drawn without replacement. Using a tree diagram, find . Give your answer as a fraction in simplest form.
/
Explanation
. .
Q11·Challenging
A bowl contains 2 apples and 3 oranges. Two pieces of fruit are selected without replacement. Using a tree diagram, find . Give your answer as a fraction in simplest form.
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Explanation
.
Q12·Challenging
A bag contains 3 white and 2 black balls. Two balls are drawn without replacement. Using a tree diagram, find . Give your answer as a fraction in simplest form.
/
Explanation
. .