A Level · Cambridge (CIE) · 9709

Correct binomial intervals, point events and probability thresholds

Include both endpoint cells, approximate a single count by a finite-width interval and check neighbouring integers for a probability target.

Section 5.5 · The normal distribution

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

Use binomial normal suitability and continuity correction.
Calculate normal interval probabilities and percentiles.

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

Quick readiness check

Which corrected interval represents 35 ≤ X ≤ 45?

What you will learn

Correct interval and exact-count binomial events.
Verify an integer threshold under the approximation and distinguish it from an exact guarantee.

Two endpoints and one-count cells

For X ~ B(100,0.40), use Y ~ N(40,24). The event 35 ≤ X ≤ 45 includes all endpoint cells, giving 34.5 < Y < 45.5. Standardised limits are approximately −1.1227 and 1.1227; the normal interval probability is approximately 0.7384.

The exact-count event X = 40 is approximated by 39.5 < Y < 40.5, not by P(Y = 40). Its standardised limits are ±0.1021 and the interval probability is approximately 0.0813. A continuous single point has probability zero, but the discrete count has a whole cell of positive area in the approximation.

Find the first valid integer under the model

To find the smallest integer k with approximate P(X ≤ k) ≥ 0.90, require Φ((k + 0.5 − 40)/√24) ≥ 0.90. Using an inverse normal calculation gives a real threshold near k = 45.78, so test k = 46. The quoted critical value 1.282 gives the same candidate.

Check both neighbours with unrounded numerical probabilities: at k = 45 the corrected cumulative probability is approximately 0.8692, below the target; at k = 46 it is approximately 0.9077, above. Thus 46 is the first integer under the normal approximation. An exact binomial target requires checking exact binomial probabilities; this approximate decision alone does not prove an exact guarantee, particularly near a target.

Pause and explain

Which event approximates X = 40 using continuity correction?

Put the idea to work

Worked example

X ~ B(100,0.40). Approximate P(X = 40).

Show the worked solution

Check the criterion and use Y ~ N(40,24).
The count-40 cell is 39.5 < Y < 40.5, with z limits ±0.5/√24.
Subtract the two cumulative areas to obtain approximately 0.0813.

Answer Approximately 0.0813.

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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

X ~ B(100,0.40). Approximate P(35 ≤ X ≤ 45).

Give me a hint

Move the lower boundary down half a unit and the upper boundary up half a unit.

Compare my reasoning

Use matching Y ~ N(40,24), with both expected counts above five.
Correct to 34.5 < Y < 45.5, giving z limits ±5.5/√24.
The interval probability is approximately 0.7384.

Approximately 0.7384; corrected interval 34.5 to 45.5.

Look for these in your work

Include both endpoint cells.
Show both standardised limits and the approximation label.

2 · Independent

Each of 80 independent trials has success probability 0.30. Let X count the successes. Use a normal approximation to find the probability of exactly 27 successes.

Give me a hint

Represent the whole count-27 cell before finding the normal area.

Compare my reasoning

Model X ~ B(80,0.30). Both np = 24 and n(1 − p) = 56 exceed five, so use Y ~ N(24,16.8), with SD √16.8.
The event X = 27 corresponds to 26.5 < Y < 27.5. Standardise both edges: (26.5 − 24)/√16.8 ≈ 0.6099 and (27.5 − 24)/√16.8 ≈ 0.8539.
Using unrounded standardised values, subtract the cumulative areas: Φ(0.8539126…) − Φ(0.6099375…) ≈ 0.0744. This is an approximate binomial probability, not a continuous point probability.

Approximately 0.0744; use 26.5 < Y < 27.5 with Y ~ N(24,16.8).

Look for these in your work

Identify the binomial model and check both expected counts.
Include the whole count-27 cell, which is above the mean of 24.
Standardise both boundaries using the standard deviation, not the variance.
Subtract cumulative areas and label the result approximate.

3 · Transfer

For X ~ B(100,0.40), find the minimum integer k giving approximate P(X ≤ k) ≥ 0.90. State the limitation for an exact target.

Give me a hint

Use k + 0.5 in the corrected boundary and check both neighbouring integers.

Compare my reasoning

Require (k + 0.5 − 40)/√24 at least the ninetieth-percentile z value, giving k near 45.78.
At k = 45 the corrected normal probability is approximately 0.8692; at k = 46 it is approximately 0.9077.
Choose k = 46 under the approximation. An exact target still requires exact binomial verification.

Minimum k = 46 under the approximation; an exact target needs exact binomial verification.

Look for these in your work

Correct the integer threshold before inverting.
Check both neighbours and preserve the approximation limitation.

Common mistakes

Correct both ends of an inclusive interval outward.
Do not replace an exact discrete count by a continuous point probability.
A threshold found by approximation is not an exact probability guarantee.

Recall without your notes

What can you explain now?

Why is P(Y = 40) not a valid approximation to P(X = 40)?

Compare with the explanation

The discrete count is represented by a unit-width cell, not a single continuous point.

Using zero-width area would erase the positive mass of that count.

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Make it stick

Map integer cells to continuous intervals, verify neighbouring threshold integers, and keep an approximate result separate from an exact binomial claim.

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.