A Level · Cambridge (CIE) · 9709

Build a discrete probability distribution from outcomes

Define the numerical variable, group elementary outcomes by its value and keep every probability in the table.

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

Enumerate equally likely elementary outcomes.
Use fractions and unordered selection counts.

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

Quick readiness check

How many heads occur in outcome HT?

What you will learn

Define a discrete random variable and its possible values.
Construct a complete probability table from a specified experiment.

The distribution of X, the number of heads in two independent fair coin tosses, groups TT at zero, HT and TH at one, and HH at two. Probabilities are 1/4, 1/2, 1/4, summing to one. Elementary strings are equally likely; the three head-count values are not.

A variable assigns a number to each outcome

A random variable X assigns a numerical value to each elementary outcome. Discrete means its possible values can be listed as a finite or countably infinite set; they need not be consecutive integers. For two independent fair coin tosses, define X as the number of heads. HH gives 2, HT and TH give 1, and TT gives 0.

The distribution table has X values 0, 1, 2 with probabilities 1/4, 1/2, 1/4. Each elementary toss string is equally likely, but the three X values are not. Check that every possible value is represented and the probabilities sum to one. Probabilities attach to values, not to the number of columns in a table.

Use the experiment’s sampling mechanism

For one fair die let Y = 1 if the result is greater than 4 and Y = 0 otherwise. Then P(Y = 0) = 4/6 = 2/3 and P(Y = 1) = 2/6 = 1/3. A variable can deliberately summarise several outcomes into one value, and its definition determines which outcomes are grouped.

Draw two balls uniformly without replacement from three red and two blue, and let X count red balls. Ten unordered pairs are equiprobable. X = 0 has one blue-blue pair, X = 1 has six mixed pairs and X = 2 has three red-red pairs. The table probabilities are 1/10, 6/10, 3/10. This table is valid although the draws are dependent; a binomial assumption is unnecessary.

Pause and explain

For two fair independent coins, what is P(X = 1 head)?

Put the idea to work

Worked example

Two independent fair coins are tossed. Build the distribution of X, the number of heads.

Show the worked solution

List TT, TH, HT and HH, each with probability 1/4.
Their X values are 0, 1, 1 and 2.
Combine equal X values to obtain probabilities 1/4, 1/2, 1/4 for 0, 1, 2.

Answer X: 0, 1, 2; probabilities: 1/4, 1/2, 1/4.

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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

Build the distribution of X, the number of heads in two independent fair coin tosses.

Give me a hint

Group the four equally likely strings by their head counts.

Compare my reasoning

TT gives 0 heads; HT and TH each give 1; HH gives 2.
The counts are 1, 2 and 1 out of four elementary outcomes.
Write probabilities 1/4, 1/2, 1/4 and check their sum is one.

X: 0, 1, 2; P(X): 1/4, 1/2, 1/4.

Look for these in your work

Define the head-count variable.
Combine both one-head outcomes.

2 · Independent

Roll one fair die. Define Y = 1 for a result greater than 4 and Y = 0 otherwise. Build its distribution.

Give me a hint

The indicator groups four die outcomes as zero and two as one.

Compare my reasoning

Results 1–4 give Y = 0.
Results 5–6 give Y = 1.
The probabilities are 4/6 = 2/3 and 2/6 = 1/3.

Y: 0, 1; probabilities: 2/3, 1/3.

Look for these in your work

Use the specified indicator definition.
Include the whole sample space exactly once.

3 · Transfer

Draw two balls uniformly without replacement from three red and two blue. Let X count red balls. Build the distribution using unordered pairs.

Give me a hint

Count all ten pairs, then split them by zero, one or two red balls.

Compare my reasoning

X = 0 has 2C2 = 1 pair; X = 1 has 3 × 2 = 6 pairs.
X = 2 has 3C2 = 3 pairs.
Divide by 5C2 = 10 to obtain 1/10, 6/10, 3/10.

X: 0, 1, 2; probabilities: 1/10, 6/10, 3/10.

Look for these in your work

Preserve the without-replacement model.
Check the three disjoint counts sum to ten.

Common mistakes

Define the variable rather than only naming the experiment.
Variable values need not be equiprobable.
Use the stated replacement mechanism when grouping outcomes.

Recall without your notes

What can you explain now?

Can a discrete variable take values −1, 0 and 2?

Compare with the explanation

Yes; discreteness does not require consecutive positive integers.

The possible values form a list, and their probabilities define the distribution.

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Make it stick

Define X, list its possible values, combine their contributing outcomes and check the complete probability table.

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.