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Probability trees practice questions with answers

Foundation and Higher · Grade 5

A second set of probability trees questions for GCSE Maths. They are different from the ones on the topic page, so work through both. Try each question, then open the answer to check your working.

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Question 1 Easy

A tree-diagram path has branches 3/4 and 2/5. What is the probability of that path?

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Answer: 3/10

  1. Multiply along path: (3/4) × (2/5) = 6/20.
  2. Simplify: 3/10.
  • Method (1): Multiply along path: (3/4) × (2/5) = 6/20.
  • Accuracy (1): Simplify: 3/10.

Question 2 Easy

A tree-diagram path has branches 1/3 and 1/3. What is the probability of that path?

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Answer: 1/9

  1. Multiply along path: (1/3) × (1/3).
  2. Result is 1/9.
  • Method (1): Multiply along path: (1/3) × (1/3).
  • Accuracy (1): Result is 1/9.

Question 3 Medium

Two counters are taken without replacement from 5 counters (2 red). Find P(both red).

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Answer: 1/10

  1. First draw: 2/5. Second draw: 1/4.
  2. Multiply: (2/5) × (1/4) = 2/20 = 1/10.
  • Method (1): First draw: 2/5. Second draw: 1/4.
  • Accuracy (1): Multiply: (2/5) × (1/4) = 2/20 = 1/10.

Question 4 Medium

With replacement, probability 1/2 then 1/2. What is the probability of both?

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Answer: 1/4

  1. Independent draws: (1/2) × (1/2).
  2. Result is 1/4.
  • Method (1): Independent draws: (1/2) × (1/2).
  • Accuracy (1): Result is 1/4.

Question 5 Hard

A branch has probability 1/4. What is the probability of not taking that branch?

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Answer: 3/4

  1. Complement: 1 - 1/4.
  2. Result is 3/4.
  • Method (1): Complement: 1 - 1/4.
  • Accuracy (1): Result is 3/4.

Question 6 Hard

A branch has probability 2/5. What is the probability of not taking that branch?

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Answer: 3/5

  1. Complement: 1 - 2/5.
  2. Result is 3/5.
  • Method (1): Complement: 1 - 2/5.
  • Accuracy (1): Result is 3/5.

Exam-style question

A longer question with marks, in the style of a written paper. This is a Yumio practice question, not a past paper question.

Exam-style question 11 marks

A branch has probability 3/5. What is the probability of not taking that branch?

Show answer

Answer: 2/5

  1. Use complement rule: P(not A) = 1 - P(A).
  2. Calculate 1 - 3/5 = 2/5.
  • Mark (1): 2/5

Quick checks

Short one-step questions. Pick an option before you open the answer.

Check 1

How do you find the probability of a combined path on a tree diagram?

  1. Multiply the probabilities along the branches
  2. Add the probabilities along the branches
  3. Divide the first by the second
Show answer

Multiply the probabilities along the branches Sequential "AND" events multiply along branches.

Check 2

What is the path probability for branches 1/2 and 1/3?

  1. 1/6
  2. 5/6
  3. 2/5
Show answer

1/6 (1/2) × (1/3) = 1/6.

Check 3

What is the path probability for branches 1/2 and 1/4?

  1. 1/8
  2. 3/4
  3. 1/6
Show answer

1/8 (1/2) × (1/4) = 1/8.

Check 4

In an experiment without replacement, what happens on the second draw?

  1. Total number of items decreases by 1
  2. Total number remains unchanged
  3. Probabilities reset
Show answer

Total number of items decreases by 1 Without replacement, one fewer item is available.

Check 5

A bag has 4 red counters out of 5. Two are drawn without replacement. What is P(both red)?

  1. 3/5
  2. 16/25
  3. 4/5
Show answer

3/5 (4/5) × (3/4) = 3/5.

Check 6

A bag has 3 red counters out of 4. Two are drawn without replacement. What is P(both red)?

  1. 1/2
  2. 9/16
  3. 3/4
Show answer

1/2 (3/4) × (2/3) = 2/4 = 1/2.

Check 7

A bag has 5 red counters out of 6. Two are drawn without replacement. What is P(both red)?

  1. 2/3
  2. 25/36
  3. 5/6
Show answer

2/3 (5/6) × (4/5) = 4/6 = 2/3.

Check 8

Two independent events each have probability 1/2. What is the probability of both occurring?

  1. 1/4
  2. 1
  3. 1/2
Show answer

1/4 (1/2) × (1/2) = 1/4.

Check 9

Two independent events each have probability 1/3. What is the probability of both occurring?

  1. 1/9
  2. 2/3
  3. 1/6
Show answer

1/9 (1/3) × (1/3) = 1/9.

Check 10

What is the probability of NOT taking a branch with probability 1/3?

  1. 2/3
  2. 1/3
  3. 1
Show answer

2/3 1 - 1/3 = 2/3.

Check 11

What is the probability of NOT taking a branch with probability 1/4?

  1. 3/4
  2. 1/4
  3. 1/2
Show answer

3/4 1 - 1/4 = 3/4.

Check 12

What is the probability of NOT taking a branch with probability 2/5?

  1. 3/5
  2. 2/5
  3. 1/5
Show answer

3/5 1 - 2/5 = 3/5.

Check 13

What do all branches radiating from a single node sum to?

  1. 1
  2. 0.5
  3. 100
Show answer

1 All mutually exclusive and exhaustive outcomes sum to 1.

Check 14

With replacement, how does the second probability compare to the first?

  1. It is identical because the sample space is restored
  2. It decreases by 1
  3. It doubles
Show answer

It is identical because the sample space is restored Replacing items preserves original proportions and independence.

Check 15

What is the probability of drawing two red counters from 5 (2 red) without replacement?

  1. 1/10
  2. 4/25
  3. 1/5
Show answer

1/10 (2/5) × (1/4) = 2/20 = 1/10.

Keep going with Probability trees

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