AQA GCSE Mathematics Paper 1 (Higher), June 2025: Question 17

4 marks · Medium difficulty · Multi-step Problem

Complete a probability tree diagram involving conditional probabilities for a bus being on time depending on whether it is raining, and calculate the probability that it is raining and the bus is not on time.

Practise this question

Question

Question 17 introduces a problem about a bus arriving at 1 pm where the probability of the bus being on time is halved when it is raining. Part (a) asks to complete a probability tree diagram with branches for 'Raining' (Yes, with given probability 1/6, and No) leading to branches for 'Bus on time' (Yes and No). Given on the diagram is the branch for 'Bus on time = No' given 'Raining = Yes' with probability 9/13. Part (b) asks to calculate the probability that at 1 pm it is raining and the bus is not on time, worth 2 marks.
Question text

17 A bus is due to arrive at a bus stop at 1pm

The probability that the bus is on time is halved when it is raining.

17 (a) Complete the tree diagram.

[2 marks]

17 (b) Work out the probability that at 1pm it is raining and the bus is not on time.

[2 marks]

Answer

Mark scheme

Show the mark scheme Mark scheme for Question 17. 17(a) awards B1 for 5/6 on 'Raining No' and 4/13 on 'Raining Yes, Bus on time Yes'; B1ft for 8/13 on 'Raining No, Bus on time Yes' and 5/13 on 'Raining No, Bus on time No'. 17(b) awards M1 for 1/6 multiplied by 9/13 and A1 for 9/78 or 3/26 or equivalent.

Q Answer Mark Comments

Raining No

and B1

Raining Yes, Bus on time Yes

8 both correct

Raining No, Bus on time Yes

13 or

17(a) and 4

ft 2 × their

5 13

Raining No, Bus on time No B1ft

13 4

and 1 – (2 × their )

where their <

13 2

Additional Guidance

oe fractions, decimals or percentages throughout

19 oe

× M1

6 13 may be seen on tree diagram

93 oe fraction, decimal or percentage

or A1

78 26

17(b)

Additional Guidance

Ignore any attempt to simplify or convert after correct answer seen

There is no follow through for incorrect values

How to answer it

Conditional Probability: Tree Diagrams & Worded Constraints

📋 What this question tests
  • Sum of probabilities: Knowing that branches emanating from a single node must always add up to 1.
  • Interpreting worded conditions: Setting up reverse relationships (e.g., if probability A is halved to get B, then B must be doubled to get A).
  • Multiplication rule: Finding combined probabilities along a specific branch path using P(A and B) = P(A) × P(B|A) .
  • Fraction multiplication: Multiplying numerators and denominators accurately without needing to simplify unless specified.
Question 17 (a) · 2 Marks

Completing the Probability Tree Diagram

Calculate missing branch values using branch sums and the given condition

--- Yes: [ 4/13 ] --- Yes (1/6) | --- No: 9/13 Start ----- | --- Yes: [ 8/13 ] --- No [ 5/6 ] --- No: [ 5/13 ]

📐 Step-by-Step Calculations

  1. Branch 1 (Raining: No):
    Branches from the first node add to 1:
    1 - 1/6 = 5/6
  2. Branch 2 (Raining Yes → On Time: Yes):
    Branches for 'Raining Yes' add to 1:
    1 - 9/13 = 4/13
  3. Branch 3 (Raining No → On Time: Yes):
    The question states punctuality is halved when raining:
    P(On time | Raining) = ½ × P(On time | Not Raining)
    Rearranging gives:
    P(On time | Not Raining) = 2 × (4/13) = 8/13
  4. Branch 4 (Raining No → On Time: No):
    Branches for 'Raining No' must sum to 1:
    1 - 8/13 = 5/13

✅ Correct Values

  • Raining (No): 5/6
  • Raining Yes, On Time (Yes): 4/13
  • Raining No, On Time (Yes): 8/13
  • Raining No, On Time (No): 5/13
Mark Scheme:
• B1: 5/6 and 4/13 placed correctly.
• B1ft: 8/13 and 5/13 (or correctly doubling their 4/13 and subtracting from 1).

❌ Common Errors

  • Halving instead of doubling: Many students see the word "halved" and calculate ½ × 4/13 = 2/13 for the sunny branch. But the bus is less likely to be on time in the rain, so it must be more likely (double) when not raining!
  • Assuming identical branches: Copying 4/13 and 9/13 straight onto the bottom set of branches without applying the condition.

🧠 Exam Technique

  • Sense-check your probabilities: Every pair of branches meeting at a point must add up to 1 ( 8/13 + 5/13 = 13/13 = 1 ).
  • Check direction of wording: "A is halved when B happens" means P(A with B) = ½ × P(A without B) . Read carefully to see which state is the base value.
Question 17 (b) · 2 Marks

Combined Probability Along a Path

Work out the probability that it is raining AND the bus is not on time

📐 Step-by-Step Calculation

  1. Trace the correct path:
    Path required: Raining (Yes) followed by Bus on time (No).
  2. Identify probabilities on this path:
    • P(Raining = Yes) = 1/6
    • P(Bus on time = No | Raining) = 9/13
  3. Multiply along the branches:
    P(Raining AND Not on time) = 1/6 × 9/13
  4. Calculate numerator and denominator:
    Numerator: 1 × 9 = 9
    Denominator: 6 × 13 = 78
    Result = 9/78 (or simplified as 3/26)

✅ Final Answer

9/78  or  3/26

Also accepts equivalent decimals (approx. 0.115) or percentages (approx. 11.5%).

Mark Scheme:
• M1: 1/6 × 9/13 (seen in working or on tree diagram).
• A1: 9/78 or 3/26 (oe).

🧠 Exam Technique & Examiner Tips

  • Do not simplify unless asked: The mark scheme explicitly notes: "Ignore any attempt to simplify or convert after correct answer seen." Leaving your answer as 9/78 gets full marks and avoids careless division errors!
  • Remember the path rule: Move across branches = multiply ( × ). Move down alternatives = add ( + ).

❌ Common Errors

  • Adding instead of multiplying: Writing 1/6 + 9/13 instead of multiplying along the path.
  • Selecting the wrong branch: Using 4/13 (bus on time) instead of 9/13 (bus not on time). Pay close attention to bold words like not in the question.

Topics

Probability · Number · 3.5 Probability · 3.1.2 Fractions, decimals and percentages

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 1 (Higher), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.