A Level · Cambridge (CIE) · 9709

Recognise when a binomial model is justified

Identify a fixed success count with independent trials and a constant success probability, and explain which assumption fails in counterexamples.

Section 5.4 · Discrete random variables

Statistics 1 · Discrete random variables · 5.4

Exam scope and assessment notes

Probability & Statistics 1 (Paper 5, AS & A Level) · Mathematics 9709, 2026–2027, section 5.4. A probability table needs non-negative probabilities summing to one. E(X) is a probability-weighted mean; Var(X) = E(X²) − [E(X)]², with squared units. A binomial model B(n,p) counts successes in a fixed non-negative integer number n of independent two-outcome trials with a constant success probability p. A geometric model Geo(p) counts the trial of the first success, starting at 1, with independent trials and a constant 0 < p ≤ 1. Formula proofs are not required. The expectation of the geometric distribution is required; its variance is not required by section 5.4. Normal approximations belong to section 5.5 and are not used here. Exact model probabilities do not establish real-world independence or constant rates without evidence. Written practice is self-checked; human subject-teacher review remains pending.

Before you begin

Distinguish independence from identical marginal probabilities.
Define an indicator or count variable.

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

Quick readiness check

What does X count in B(n,p)?

What you will learn

State the four binomial modelling assumptions.
Use B(n,p) notation and reject unsuitable trial mechanisms.

Four fixed independent trials share success probability 1/3. The illustrated S F S F sequence has two successes and probability (1/3)²(2/3)² = 4/81. There are six choices for the two success positions, so P(X = 2) = 6 × 4/81 = 8/27 for X ~ B(4,1/3). X counts successes rather than the first-success trial.

Count successes in a fixed experiment

X ~ B(n,p) means X counts successes in n independent trials with the same success probability p, with each trial classified as success or failure. n is a fixed non-negative integer and 0 ≤ p ≤ 1. The support is 0, 1, …, n; X records how many successes, rather than the trial on which the first success occurs.

For four stipulated independent trials with success probability 1/3 each, X ~ B(4,1/3). Calling an outcome ‘success’ is a mathematical classification: a chosen die result, a correct answer or a flagged item can be designated success. Define the event and justify the assumptions from the supplied mechanism, rather than assuming real students’ answers are independent and equally difficult.

A count alone is not enough

Drawing two balls without replacement from three red and two blue, with red called success, does not produce a binomial model: the later probability changes with the first draw and the trials are dependent. It still has a valid discrete distribution, which can be obtained by enumerating pairs or using a tree.

Three independent trials with success probabilities 1/2, 2/3 and 3/4 are not binomial because their probabilities are not constant. Independence alone is insufficient. A variable counting trials until success instead has a changing stopping time, so it does not have a fixed n; it may fit a geometric model under its own assumptions. Do not force a named model when its defining conditions fail.

Pause and explain

Two draws without replacement from three red and two blue: is the red count binomial?

Put the idea to work

Worked example

Four independent trials each have success probability 1/3. Define X and its distribution.

Show the worked solution

Let X be the number of successful trials.
There are four fixed trials, two outcome classifications, independence and common p = 1/3.
Write X ~ B(4,1/3), with values 0–4.

Answer X ~ B(4,1/3); X counts successes, with support 0, 1, 2, 3, 4.

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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

A stipulated model has ten independent trials, each with success probability 0.20. X counts successes. State the distribution and support.

Give me a hint

Check fixed count, two classifications, independence and common probability.

Compare my reasoning

There are ten fixed trials with success/failure outcomes.
Independence and constant p = 0.20 are supplied.
Write X ~ B(10,0.20), with X = 0, 1, …, 10.

X ~ B(10,0.20); support 0–10.

Look for these in your work

Define X as a success count.
Identify all required assumptions.

2 · Independent

Draw two balls without replacement from three red and two blue, counting red as success. Assess a binomial model.

Give me a hint

Compare the second red probability after a red with its probability after a blue.

Compare my reasoning

After red, second red is 2/4; after blue, it is 3/4.
The trial probabilities depend on the first result and are not constant.
The red count is not binomial; use the without-replacement distribution instead.

Not binomial: dependence and changing success probability violate the model.

Look for these in your work

Give specific conditional probabilities.
Identify the failed assumptions rather than just naming the alternative.

3 · Transfer

Three trials are independent but have success probabilities 1/2, 2/3 and 3/4. X counts successes. Is X binomial? Find the all-success probability without assuming a binomial model.

Give me a hint

Independence allows a product but does not make the probabilities identical.

Compare my reasoning

The constant-p requirement fails, so X is not binomial.
Multiply the supplied independent success probabilities.
P(all succeed) = (1/2)(2/3)(3/4) = 1/4.

Not binomial; all-success probability 1/4.

Look for these in your work

Separate independence from a common probability.
Use each trial’s actual probability in the product.

Common mistakes

A success count needs all four model assumptions.
Without replacement can destroy both independence and constant probability.
A waiting-time variable has a different definition from a fixed-trial count.

Recall without your notes

What can you explain now?

Are independent trials with different success probabilities binomial?

Compare with the explanation

No; the constant-p condition fails.

Independence and identical success probability are separate assumptions.

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Make it stick

Define the count and check fixed n, two classifications, independence and constant p before writing B(n,p).

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.