A Level · Cambridge (CIE) · 9709

Check when a normal approximation to binomial is suitable

Test expected successes and failures, preserve binomial assumptions and use the correct matching normal mean and variance.

Statistics 1 · Normal distribution · 5.5

Exam scope and assessment notes

Probability & Statistics 1 (Paper 5, AS & A Level) · Mathematics 9709, 2026–2027, section 5.5. N(μ,σ²) uses variance as its second parameter, with σ > 0; probabilities are areas, not density heights. Show the full standardisation Z = (X − μ)/σ. The MF19 table gives Φ(z) = P(Z ≤ z), with negative values found by symmetry. Model sketches and shaded probabilities may be required. Probability answers use quoted four-decimal table values where supplied; calculations with non-tabulated z use unrounded standardisation and numerical normal probabilities, so table interpolation can differ slightly in the last digits. Inverse examples use the quoted three-decimal MF19 critical values and are approximate. A binomial normal approximation requires both np > 5 and n(1 − p) > 5, and a continuity correction. It remains approximate, including for integer probability thresholds. A continuous normal variable has zero probability at a single exact value; a discrete binomial count does not. Sample means, the Central Limit Theorem and hypothesis testing belong to Statistics 2 and are not covered here. Model assumptions need evidence in real applications. Written practice is self-checked; human subject-teacher review remains pending.

Before you begin

Identify binomial modelling assumptions.
Calculate binomial mean and variance.

Work on paper. Try each question before revealing help, and explain why your method works.

Quick readiness check

For B(100,0.40), what are np and n(1 − p)?

What you will learn

Check np > 5 and n(1 − p) > 5 for the syllabus approximation.
Match B(n,p) with the approximate N(np,np(1 − p)) model.

A binomial distribution B(100,0.40) has expected successes 40 and failures 60, both greater than 5. Its matching normal approximation is N(40,24), with standard deviation square root of 24. Binomial cells for counts 23 through 57 are shown with their probability as area, after changing to the common standardised horizontal scale; the smooth standard normal density overlays them. Only part of each distribution is drawn. The approximation still needs continuity correction and is not exact.

Test both sides of the trial outcome

Cambridge section 5.5 requires n to be sufficiently large that both np > 5 and nq > 5, with q = 1 − p. For X ~ B(100,0.40), expected successes are 40 and expected failures are 60, so the criterion is met. A suitable matching normal variable is Y ~ N(40,24), with SD √24.

The binomial experiment still needs fixed n, independence, two classifications and a constant p. The normal approximation does not establish those assumptions. It approximates the distribution of the success count; it does not convert a dependent or changing-probability experiment into a binomial one.

A large n alone is insufficient

For B(200,0.01), np = 2 and nq = 198: the small success expectation fails the criterion despite 200 trials. For B(50,0.10), np = 5, which does not satisfy the stated strict inequality np > 5. Keep that boundary explicit rather than replacing it with a different rule of thumb.

For B(40,0.50), both expected counts are 20, and the matching normal is N(20,10), with SD √10. Meeting the criterion makes the approximation available under the syllabus, but does not guarantee negligible error for every tail. Use a continuity correction, label the answer approximate and consider exact binomial calculation when an exact decision threshold matters.

Pause and explain

Does B(50,0.10) satisfy both stated approximation inequalities?

Put the idea to work

Worked example

X ~ B(100,0.40). Check the normal-approximation criterion and state the matching model.

Show the worked solution

np = 40 and n(1 − p) = 60, both above five.
Mean = np = 40 and variance = np(1 − p) = 24.
Use Y ~ N(40,24) as an approximation, with continuity correction when translating events.

Answer Criterion met; Y ~ N(40,24) approximately matches X.

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From guided practice to a new situation

Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.

1 · Guided

Check the normal approximation for B(100,0.40) and state its matching normal mean, variance and SD.

Give me a hint

Check both expected counts before using the moment formulas.

Compare my reasoning

np = 40 and nq = 60, both greater than five.
The matching mean is 40 and variance is 100(0.4)(0.6) = 24.
Use approximate N(40,24), with SD √24 and event continuity corrections.

Criterion met; approximate N(40,24), SD √24.

Look for these in your work

Check both strict inequalities.
Distinguish variance from SD and preserve the approximation label.

2 · Independent

Assess a normal approximation for B(50,0.10) using the stated Cambridge criterion.

Give me a hint

Compare np with a strict greater-than-five condition.

Compare my reasoning

np = 5 and nq = 45.
The failure expectation exceeds five, but the success expectation equals it.
The stated criterion is not met; do not assert a normal approximation is justified by this rule.

Criterion not met: np = 5 is not > 5.

Look for these in your work

Apply the strict boundary correctly.
Check both outcome expectations.

3 · Transfer

Compare B(200,0.01) and B(40,0.50) for normal approximation; state the matching model where justified.

Give me a hint

A larger n may still have a small expected success count.

Compare my reasoning

For B(200,0.01), np = 2 and nq = 198, so the criterion fails.
For B(40,0.50), np = nq = 20, so the criterion is met.
The latter matching model is approximate N(20,10), with SD √10.

B(200,0.01) fails; B(40,0.50) meets the criterion and matches N(20,10) approximately.

Look for these in your work

Do not decide from n alone.
Use the matching binomial mean and variance.

Common mistakes

Test expected failures as well as successes.
The syllabus criterion uses strict greater-than-five boundaries.
Use np(1 − p) as variance and its square root as SD.

Recall without your notes

What can you explain now?

Does n = 200 guarantee a useful normal approximation if p = 0.01?

Compare with the explanation

No; np = 2 fails the criterion.

The distribution may remain strongly skewed when successes are rare.

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Make it stick

Verify the binomial assumptions and both expected counts, then use N(np,npq) with an explicit approximation and continuity correction.

If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.

Keep learning Mathematics

Check the official syllabus 2026–2027

Course scope and assessment

312 sequenced lessons provide 936 guided, independent and application problems covering the 160 objectives in sections 1.1–1.8, 2.1–2.6, 3.1–3.9, 4.1–4.5, 5.1–5.5 and 6.1–6.5. The map lists all 38 sections across six alternative components; all eight Pure Mathematics 1 sections and all six Pure Mathematics 2 sections, from Paper 2 Algebra through Numerical solution of equations, currently have full teaching sequences. Paper 3 Algebra, section 3.1, links the six shared foundation lessons plus six lessons on partial fractions and rational binomial expansions; Paper 3 logarithms, trigonometry and numerical equations use the shared lessons, with the Paper 1 graph foundation linked into trigonometry. Paper 3 differentiation combines the shared methods with its inverse-tangent derivative lesson. Paper 3 Integration links four shared standard and identity lessons plus seven advanced lessons, covering all six section 3.5 requirements. Eight Vectors lessons cover all six section 3.7 requirements, with 2D and 3D line and solid applications. Nine Differential equations lessons cover all four section 3.8 requirements, including Paper 3 integration techniques, initial conditions, equilibrium solutions and conditional model interpretations. Eleven Complex numbers lessons cover all eight section 3.9 requirements, with seven Argand diagrams and explicit argument and boundary checks. All nine Pure Mathematics 3 sections now have teaching linked to every mapped requirement. Nine Forces and equilibrium lessons cover all seven section 4.1 requirements, with six force diagrams, signed incline-friction intervals, reaction-dependent pull and push limits and separate-body third-law checks. Nine Kinematics lessons cover the four official section 4.2 requirements, with the graph requirement mapped to three relationships. Six motion diagrams support directed quantities, graph slopes and signed areas, Paper 1 time-calculus methods, reversal distances, constant-acceleration stages and shared-clock particle meetings. Six Momentum lessons cover both section 4.3 requirements, including signed system totals, conservation assumptions, outgoing velocities, coalescence, unknown masses and successive impacts. Four diagrams show system momentum through the stipulated events, and supplied kinetic-energy checks distinguish conservation of momentum from changes in kinetic energy. Twelve Newton’s laws lessons cover all four section 4.4 requirements, with vertical support and release stages, rough horizontal and incline motion, inferred parameters, hanging and surface pulley systems, rigid tow-bar thrust and slack-string transitions. Seven force diagrams support the separate-particle balances. Nine Energy, work and power lessons cover all five section 4.5 requirements, with angle-dependent work, kinetic and reference potential energy, signed net-work balances, smooth curved slides, rough and driven slopes, ideal connected-system energy and instantaneous power on hills. Five diagrams distinguish the energy inputs, losses and valid positive-speed force model. All five Mechanics sections now have teaching and staged practice linked to their mapped requirements. Thirteen Representation of data lessons cover all five section 5.1 requirements, including choice and limitations of displays, ordered and back-to-back stems, raw summaries and box plots, unequal-width density histograms, cumulative-frequency interpolation, descriptive moment calculations, grouped estimates, coding, combining two datasets and correcting totals. Six accessible diagrams retain repeated observations and distinguish numerical spread, frequency area and estimated readings. Ten Permutations and combinations lessons cover both official section 5.2 requirements, with selection and ordered roles, repeated objects, adjacency blocks, non-adjacency gaps, leading-zero cases, labelled multi-row seating and mixed restrictions. Six accessible diagrams show outcome trees, ordered versus unordered choices, repeated copies, blocks, gaps and distinct row seats. Eleven Probability lessons cover all four section 5.3 requirements, with equiprobable elementary outcomes, selection-based probabilities, complements and disjoint regions, independence tests, restricted sample spaces, two-way tables, replacement and removal trees, reversed conditions and unknown parameters. Seven accessible diagrams show dice outcomes, random subsets, event regions, a conditioned grid, a count table and two probability trees. Twelve Discrete random variables lessons cover all three official section 5.4 requirements, with complete probability tables, moment and parameter checks, binomial assumptions, exact probabilities and moments, geometric first-success probabilities, tails, expectation and integer reliability thresholds. Six accessible diagrams distinguish outcome grouping, weighted moments, fixed trials, discrete probability bars, waiting conventions and integer tail deadlines. Twelve Normal distribution lessons cover all three official section 5.5 requirements, with continuous-area interpretation, MF19 cumulative tables, full standardisation, tail and interval regions, conditional probabilities, percentiles, one- and two-parameter inference, binomial suitability, continuity corrections and approximate integer thresholds. Seven accessible diagrams show model areas, table reading, aligned scales, intervals, quantiles, matched binomial and normal shapes and whole integer cells. All five Statistics 1 sections now have teaching and staged practice linked to their mapped requirements. Fourteen Poisson distribution lessons cover all five official section 6.1 requirements, with homogeneous event-count assumptions, exposure scaling, exact masses and tails, conditional count events, moments, parameter inference, probability duration targets, disjoint and overlapping windows, rare-count binomial approximations, matched normal approximations and continuity corrections. Nine accessible diagrams show event exposures, unit conversions, discrete bars, complete tail complements, population moments, shared windows, approximate count models and integer cells. Exact checks distinguish approximate and exact close-target decisions. Fourteen Linear combinations lessons cover the six listed results in official section 6.2, with affine moments, signed scales and constant cases, expectation without independence, independent squared variance contributions, fixed and random fees, independent copies versus reused values, affine normal events, independent normal sums and differences, weighted and conditional areas, coefficient inference, exact independent Poisson sums and overlapping exposure limits. Nine accessible diagrams distinguish scale changes, variance contributions, component costs, copy mechanisms, aligned normal scales, summed areas, Poisson count pairs and unsupported distribution claims. Fourteen Continuous random variables lessons cover both official section 6.3 requirements, with density validity and units, support-aware intervals, complements and conditional regions, first and second moments, percentiles by direct density areas, symmetry checks, parameter inference, infinite-support power and exponential integrals, divergent-moment limits and combined applications. Nine accessible diagrams distinguish density height from probability, normalisation, negative-density failures, central and conditional areas, mean versus median, percentile locations and partial infinite tails. Explicit cumulative distribution function knowledge is not required. Eighteen Sampling and estimation lessons cover all eight official section 6.4 requirements, with population and frame definitions, random-number selection, sampling-quality limits, independent sample-mean moments, normal means, CLT conditions, SE scaling, raw, summarised and coded unbiased estimates, known-variance and large-sample mean intervals, repeated-sampling interpretation, precision planning, proportion intervals, sparse-outcome limits and contextual applications. Ten accessible diagrams distinguish selection frames, number rejection, repeated sample means, population versus mean density, CLT models, variance divisors, interval targets, illustrative coverage, integer size checks and proportion margins. Eighteen Hypothesis tests lessons cover all five official section 6.5 requirements, with pre-specified alternatives, inclusive count tails, explicit equal-tail conventions, integer critical regions and actual significance, exposure-aware Poisson tests, continuity-corrected count approximations, normal and large-sample mean tests, confidence-interval connections, error classifications, fixed-region binomial, Poisson and normal error probabilities and sample-size power comparisons. Nine accessible diagrams show the testing procedure, exact count regions, unequal Poisson tails, discrete significance jumps, corrected count boundaries, matching mean intervals and alternative error distributions. All shared lessons are counted once in the course total. All 38 sections now link teaching and practice. Eight earlier overview lessons remain available, but do not count as full section coverage. This is a content-availability measure, not full programme completion or exam readiness. Written practice is self-checked, not officially marked. Check the 2028–2030 syllabus separately if sitting an examination from 2028; this teaching map uses 2026–2027 requirements.