Unit 4 · Probability & Statistics 1 (Paper 5)
The Complete Cambridge International AS & A Level Mathematics (9709) Guide · Cambridge International AS & A Level Mathematics (9709)Unit 4 · Probability & Statistics 1 (Paper 5)
- The Complete Cambridge International AS & A Level Mathematics (9709) Guide
- Cambridge International AS & A Level Mathematics (9709)
- 5 sections
Cambridge International does not publish a per-unit weighting for this exam. Exam structure (official): AS Level: Pure Mathematics 1 (110 min, 60%) · Pure Mathematics 2 (75 min, 40%) · Mechanics (75 min, 40%) · Probability & Statistics 1 (75 min, 40%); A Level: Pure Mathematics 1 (110 min, 30%) · Pure Mathematics 3 (110 min, 30%) · Mechanics (75 min, 20%) · Probability & Statistics 1 (75 min, 20%) · Probability & Statistics 2 (75 min, 20%). Be ready to carry the same unit skills and representations into a complete answer on every paper. Correct endpoints, widths, and model parameters protect the calculation.
- How Cambridge International AS & A Level Mathematics (9709) assesses this Cambridge International publishes no per-unit percentage for Cambridge International A Level Mathematics; Pure Mathematics 1 accounts for 60% of the AS Level total and 30% of the A Level total. · AS Level: Pure Mathematics 1 (110 min, 60%) · Pure Mathematics 2 (75 min, 40%) · Mechanics (75 min, 40%) · Probability & Statistics 1 (75 min, 40%) | A Level: Pure Mathematics 1 (110 min, 30%) · Pure Mathematics 3 (110 min, 30%) · Mechanics (75 min, 20%) · Probability & Statistics 1 (75 min, 20%) · Probability & Statistics 2 (75 min, 20%) · show the model with grouped frequency table or histogram with class widths, tree diagram, event notation, or probability table, discrete distribution table and normal curve with a corrected boundary
- Key skills Representation of Data: Diagrams, Measures of Central Tendency and Variation, Permutations and Combinations: Arrangements and Selections with Restrictions, Probability: Addition and Multiplication Rules, Independence and Conditional Probability, Discrete Random Variables: Expectation, Variance, Binomial and Geometric Distributions, The Normal Distribution: Standardising and the Normal Approximation to the Binomial
- How to study for Unit 4 This page turns grouped frequency table or histogram with class widths, tree diagram, event notation, or probability table, discrete distribution table and normal curve with a corrected boundary into one route: Define the event or model parameters and check class widths, replacement, support, and endpoint wording..
- The organizing decision Identify the data display, counting structure, event dependence, or probability distribution before evaluating the requested quantity.
What Cambridge International AS & A Level Mathematics (9709) Unit 4 covers
Use this map to connect each assessed skill to the relationship or representation that makes it visible.
Representation of Data: Diagrams, Measures of Central Tendency and Variation
frequency density; permutation; conditional probability02Permutations and Combinations: Arrangements and Selections with Restrictions
conditional probability; independence; expectation03Probability: Addition and Multiplication Rules, Independence and Conditional Probability
expectation; binomial model; normal approximation04Discrete Random Variables: Expectation, Variance, Binomial and Geometric Distributions
normal approximation; continuity correction; frequency density05The Normal Distribution: Standardising and the Normal Approximation to the Binomial
frequency density; permutation; conditional probabilityCambridge International does not publish a per-unit weighting for this exam.
Official unit name and weighting: Cambridge International syllabus.
Probability and data
Connect the published share to the unit model
Correct endpoints, widths, and model parameters protect the calculation.
For probability and data, the following pages use newly written examples and data. The probability and data scoring comparisons are learning aids based on the published syllabus and the general requirement to show necessary working, not official mark schemes.
Before work on probability and data, identify the domain, units, accuracy instruction, and event boundary. Preserve exact values in probability and data until a final approximation is requested.
The decision that organizes this unit
Define the system and choose the route before calculating
Identify the data display, counting structure, event dependence, or probability distribution before evaluating the requested quantity.
Define the event or model parameters and check class widths, replacement, support, and endpoint wording.
Mechanism route and repair branches
Relationships to preserve
- Histogram height is frequency density, so unequal class widths must be included.
- Without replacement changes later conditional probabilities and therefore affects independence.
- A discrete event and its normal approximation must describe the same endpoints, including any continuity correction.
Representations to read
- grouped frequency table or histogram with class widths
- tree diagram, event notation, or probability table
- discrete distribution table and normal curve with a corrected boundary
Branches to reject
- using frequency rather than frequency density for unequal classes
- using replacement probabilities for a draw without replacement
- taking the wrong normal tail or shifting an inclusive boundary in the wrong direction
| Key concept | Why it's hard | What scores |
|---|---|---|
| Data display | Unequal widths change histogram height | Uses density and contextual comparison |
| Event structure | Replacement changes later probabilities | Defines intersections and conditions |
| Distribution model | Tail endpoints can be off by one | Names parameters and uses correction when needed |
How Cambridge International AS & A Level Mathematics (9709) assesses Probability & Statistics 1 (Paper 5)
What a complete response must make visible
Match the task to evidence that a reader can audit, then check the most likely reasoning failure before finalizing the response.
| Task | Evidence to show | Hurdle |
|---|---|---|
| Representation of Data: Diagrams, Measures of Central Tendency and Variation | frequency density; permutation; conditional probability | using frequency rather than frequency density for unequal classes |
| Permutations and Combinations: Arrangements and Selections with Restrictions | conditional probability; independence; expectation | using replacement probabilities for a draw without replacement |
| Probability: Addition and Multiplication Rules, Independence and Conditional Probability | expectation; binomial model; normal approximation | taking the wrong normal tail or shifting an inclusive boundary in the wrong direction |
| Discrete Random Variables: Expectation, Variance, Binomial and Geometric Distributions | normal approximation; continuity correction; frequency density | using frequency rather than frequency density for unequal classes |
| The Normal Distribution: Standardising and the Normal Approximation to the Binomial | frequency density; permutation; conditional probability | using replacement probabilities for a draw without replacement |
Grouped data and arrangements
Read each question as it appears on the exam paper, choose your answer, then tap it to see the feedback.
Practice 1: Grouped data and arrangements
The table shows the frequencies for three classes.
The intervals [0, 10), [10, 20) and [20, 40) have frequencies 8, 14 and 18 respectively. (a) Find the frequency density for [20, 40). (b) Find the number of arrangements of the letters of MATHS in which the vowels are together.
[4]Show all necessary working.
A. 18 ÷ 20 = 0.9, and the only vowel is A, so all 5! = 120 arrangements satisfy the condition.
✅ Correct! 😊 Correct. Frequency density uses the width 20, and a condition that the sole vowel is together introduces no restriction.
B. 18 ÷ 40 = 0.45, and treating the vowels as one block gives 4! = 24 arrangements.
❌ Not quite 😭 This uses the upper boundary instead of the class width and invents a multi-letter vowel block even though MATHS contains only A.
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C. The frequency density is 18, and there are 5! ÷ 2! = 60 arrangements.
❌ Not quite 😭 This ignores unequal class widths and divides for a repeated letter, but every letter in MATHS is distinct.
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D. 18 ÷ 20 = 0.9, and there are 2 × 4! = 48 arrangements because the vowel block has two orders.
❌ Not quite 😭 The density is correct, but a one-letter block has only one internal order. The factor of 2 has no basis.
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Full explanation
- Step 1The width of [20, 40) is 40 − 20 = 20.
- Step 2Frequency density is frequency divided by class width, so 18 ÷ 20 = 0.9.
- Step 3MATHS has five distinct letters and only one vowel, A, so the condition does not restrict its position.
- Step 4All permutations are allowed, giving 5! = 120.
Practice 2: Sampling without replacement
A bag contains 5 red counters and 3 blue counters.
Two counters are selected at random without replacement. Find the probability that one counter of each color is selected.
[4]Show all necessary working.
A. (5/8)(3/7) + (3/8)(5/7)
✅ Correct! 😊 Correct. The two disjoint orders are red then blue and blue then red, and the second denominator is 7 because there is no replacement.
B. (5/8)(3/8) + (3/8)(5/8)
❌ Not quite 😭 This keeps the original denominator for the second draw, so it models sampling with replacement rather than without replacement.
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C. (5/8)(3/7)
❌ Not quite 😭 This counts only red followed by blue. It omits the equally valid order blue followed by red.
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D. (5/8) + (3/8)
❌ Not quite 😭 This adds the probabilities of the first counter's colors and therefore describes a certain first draw, not the two-draw event.
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Full explanation
- Step 1Split the event into the disjoint orders red then blue and blue then red.
- Step 2Without replacement, these probabilities are (5/8)(3/7) and (3/8)(5/7).
- Step 3Add the two orders: 15/56 + 15/56 = 30/56.
- Step 4Therefore the required probability is 15/28.
Key terms for Unit 4: Probability & Statistics 1 (Paper 5)
Models, uses, and boundaries
- Conditional probability is written as the intersection probability divided by the conditioning probability
- Divide frequency by class width for the histogram and count arrangements only after checking the actual restriction. Conditional probability is written as the intersection probability divided by the conditioning probability.
- Frequency density fixes the histogram height, while the permutation expression counts the arrangements
- Multiply the changing without-replacement probabilities and compare a conditional probability with the corresponding marginal probability. Frequency density fixes the histogram height, while the permutation expression counts the arrangements.
- The without-replacement probability is followed by the unequal probabilities used to test independence
- Normalize the discrete probability table, form the expectation, and use the binomial endpoint probability. The without-replacement probability is followed by the unequal probabilities used to test independence.
- The binomial point probability and expectation summarise the discrete model used here
- Calculate the binomial mean and standard deviation, apply the correct continuity boundary, and take the required normal tail. The binomial point probability and expectation summarise the discrete model used here.
- The continuity-corrected boundary is standardised before the normal probability is read
- Write conditional probability as the intersection probability divided by the probability of the stated condition. The continuity-corrected boundary is standardised before the normal probability is read.
Cambridge International AS & A Level Mathematics (9709) Unit 4 FAQ
How much of Cambridge International AS & A Level Mathematics (9709) does Unit 4 carry?
Cambridge International does not publish a per-unit weighting for this exam.
What is the first move on a Probability & Statistics 1 (Paper 5) problem?
Define the event or model parameters and check class widths, replacement, support, and endpoint wording.
Which relationships should I preserve?
Histogram height is frequency density, so unequal class widths must be included. Without replacement changes later conditional probabilities and therefore affects independence. A discrete event and its normal approximation must describe the same endpoints, including any continuity correction.
Which representations should I practice?
Practice moving among grouped frequency table or histogram with class widths, tree diagram, event notation, or probability table, discrete distribution table and normal curve with a corrected boundary.
What error should I check before submitting an answer?
Check for using frequency rather than frequency density for unequal classes; using replacement probabilities for a draw without replacement; taking the wrong normal tail or shifting an inclusive boundary in the wrong direction.
Evidence workshop
Continue from the free model into complete practice
The full unit guide continues with the chapter’s worked examples, figures, scoring tables, and answer checks.
- Grouped data and arrangements
- Conditional probability
- Discrete models
- Normal approximation
Full unit practice. Open the complete guide for the full evidence workshop and synthesis.
Related Cambridge International AS & A Level Mathematics (9709) unit guides
Cambridge International AS & A Level Mathematics (9709) Exam Guide & Review
The whole exam and its official unit sequence.01Pure Mathematics 1 (Paper 1)
Cambridge International does not publish a per-unit weighting for this exam.02Pure Mathematics 3 (Paper 3, A Level)
Cambridge International does not publish a per-unit weighting for this exam.03Mechanics (Paper 4)
Cambridge International does not publish a per-unit weighting for this exam.05Probability & Statistics 2 (Paper 6, A Level)
Cambridge International does not publish a per-unit weighting for this exam.How to study Cambridge International AS & A Level Mathematics (9709) Unit 4
Start with the organizing decision
Before solving, restate the decision in operational terms: Identify the data display, counting structure, event dependence, or probability distribution before evaluating the requested quantity.. Your first written move should be to Define the event or model parameters and check class widths, replacement, support, and endpoint wording..
Practice the same idea in several representations
Rotate through grouped frequency table or histogram with class widths, tree diagram, event notation, or probability table, discrete distribution table and normal curve with a corrected boundary. Use each representation to practice Representation of Data: Diagrams, Measures of Central Tendency and Variation, Permutations and Combinations: Arrangements and Selections with Restrictions, Probability: Addition and Multiplication Rules, Independence and Conditional Probability, Discrete Random Variables: Expectation, Variance, Binomial and Geometric Distributions, The Normal Distribution: Standardising and the Normal Approximation to the Binomial, and explain what stays invariant when the surface form changes.
Turn each error into a repair check
After every attempt, audit the response for using frequency rather than frequency density for unequal classes; using replacement probabilities for a draw without replacement; taking the wrong normal tail or shifting an inclusive boundary in the wrong direction. Then redo only the first step that made the reasoning diverge, keeping units, direction, and model conditions visible.
Confirm current course details in the official Cambridge International syllabus for the 2026-2027 administration.
Where to go next
Return to the Cambridge International AS & A Level Mathematics (9709) to see every unit and plan your revision.