A Level · Cambridge (CIE) · 9709

Define geometric waiting times starting at the first trial

Count the trial of the first success from 1 and distinguish it from the number of failures before success.

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

Identify success and failure outcomes.
Recognise independence and conditional probabilities.

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

Quick readiness check

For outcomes failure, failure, success, what is the first-success trial?

What you will learn

Define the Cambridge geometric variable and its support.
Check independence and constant probability in a stopping experiment.

For independent trials with constant success probability 1/4, the path failure, failure, success ends at trial three. The first-success variable T equals 3 while the failure count is T − 1 = 2. The path probability is (3/4)²(1/4) = 9/64. A geometric first-success variable begins at trial 1.

The random variable is the trial number

T ~ Geo(p) counts the trial on which the first success occurs. Trials are independent and each has the same success probability p, with 0 < p ≤ 1. The possible values are 1, 2, 3, …, because success may occur on the very first trial. Cambridge section 5.4 uses this convention.

If the observed sequence is failure, failure, success, then T = 3, while the number of failures before success is T − 1 = 2. These variables have different supports. A formula for the failure count cannot be copied unchanged onto T. Define which variable an event describes before using a calculator or another textbook’s convention.

A stopping rule does not guarantee a geometric model

Rolling a fair die independently until the first six gives T ~ Geo(1/6). The number of rolls is random and the first-six probability remains 1/6 on each roll. This contrasts with a binomial experiment, where the total number of trials is fixed in advance and the variable counts successes.

Drawing without replacement from two red and three blue until the first red is not geometric: removal changes conditional probabilities, creates dependence and limits the first-red trial to at most 4. At p = 1, T = 1 with certainty. At p = 0, no finite first-success trial exists, so it does not define the geometric variable used here. The infinite support for 0 < p < 1 is a model property, not a promise about real-world trial conditions staying constant forever.

Pause and explain

Independent fair die rolls stop at the first six. Which model describes its trial number T?

Put the idea to work

Worked example

Independent trials each have success probability 1/4. The first outcomes are failure, failure, success. State T and the failure count.

Show the worked solution

T counts the trial of the first success, starting at one.
The first success occurs on trial three, so T = 3.
There were T − 1 = 2 failures beforehand.

Answer T = 3; failures before success = 2.

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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

Trials are independent with success probability 1/4. Outcomes are failure, failure, success. State T and the number of failures before success.

Give me a hint

Include the successful trial in T, then subtract one for failures.

Compare my reasoning

The first success is on trial three.
Thus T = 3 under the first-trial convention.
The preceding failure count is T − 1 = 2.

T = 3; two failures before success.

Look for these in your work

State the first-success convention.
Keep the two variable definitions distinct.

2 · Independent

A fair die is rolled independently until the first six. Define T, its distribution and support.

Give me a hint

The common probability is that of a six on one roll.

Compare my reasoning

Let T be the trial number of the first six.
Each independent roll has p = 1/6.
Write T ~ Geo(1/6), with support 1, 2, 3, … .

T ~ Geo(1/6); support 1, 2, 3, … .

Look for these in your work

Define the waiting variable explicitly.
Explain the constant probability and independence.

3 · Transfer

Draw without replacement from two red and three blue until the first red. Is its first-red trial geometric? State the possible trial numbers.

Give me a hint

Compare conditional probabilities after removals and the latest possible first red.

Compare my reasoning

Initially red has probability 2/5; after one blue it becomes 2/4.
Removal changes probabilities and makes trials dependent, so the model is not geometric.
The first red can occur on trials 1, 2, 3 or 4; three blues are the maximum preceding failures.

Not geometric; possible first-red trials are 1–4.

Look for these in your work

Identify the failed assumptions using the mechanism.
Distinguish finite support from geometric support.

Common mistakes

The first-success variable starts at 1.
Failure count and first-success trial differ by one.
Without replacement does not supply a constant independent success probability.

Recall without your notes

What can you explain now?

What value does T take if the first trial succeeds?

Compare with the explanation

T = 1.

The variable counts the successful trial itself, not just preceding failures.

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Make it stick

For Geo(p), define T as the first-success trial beginning at 1, then check independent trials and constant positive 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.