AS91586 · Probability Distributions Bench

Module 01 · The brief

What the standard actually asks for

Apply probability distributions in solving problems. Four credits, examined end of year on printed paper alongside 91584 and 91585. Bring a graphing calculator — the specifications warn that candidates without one are disadvantaged. A Formulae and Tables Booklet is supplied.

The three method strands

Straight from the standard's explanatory notes.

Discrete and continuous probability distributions.
Mean and standard deviation of random variables.
Distributions of true probabilities vs model estimates vs experimental estimates.

Strand 3 is where most Excellence marks live and it is the strand most courses under-teach. Budget a fortnight for it.

How a question is graded

Three questions, each graded holistically N0–E8.

Band	Requires
uAchieved	Method selected and used correctly
rMerit	Relational — steps linked, findings related to context
tExcellence	Extended abstract — strategy, chain of reasoning, generalisation

Grade cut scores (2025): N 0–7 · A 8–13 · M 14–18 · E 19–24

The five distributions you must cover

Named explicitly in the assessment specifications.

Examiner house rules

Name the distribution and its parameters. Schedules repeatedly read "probability correct AND distribution and parameters stated".
State assumptions, in context, every time.
Questions sit in real-life contexts and answers must be interpreted in context.

Sensible rounding expected; early rounding may be penalised.
Minor errors are not penalised — keep going after a slip.
Solutions may draw on knowledge up to Statistics Curriculum Level 7.

Module 02 · Foundations

Random variables and expectation algebra

Teach this before any named distribution. It is the machinery underneath all five, and it is where the abstract Excellence questions hide.

Discrete random variable calculator

Edit the table. Everything below recomputes live, with the working shown the way a marker wants to see it.

Linear transformation: Y = aX + b

Uses the table above.

a

b

The trap: nX vs X₁+…+Xₙ

A reliable Excellence discriminator. Same mean, different spread.

n (independent copies / multiplier)

Rules to have automatic

E(aX + b) = aE(X) + b VAR(aX + b) = a²·VAR(X) E(X ± Y) = E(X) ± E(Y) VAR(X ± Y) = VAR(X) + VAR(Y) (X, Y independent — note the +) VAR(X) = E(X²) − [E(X)]² SD(X) = √VAR(X)

Module 03 · The bench

Distribution bench

Pick a model, set its parameters, ask a probability question. The readout writes the answer in the form the marking schedule expects — distribution named, parameters stated, working shown.

Probability question

Conditions and vocabulary for this model

Module 04 · Working backwards

Solving for the parameter

Merit questions rarely hand you the parameters. They hand you a probability and make you recover λ, σ, μ, n or p. These are the five inverse patterns that keep recurring.

Module 05 · True vs model vs experimental

Model fit bench

Fit a triangular or uniform model to real sample data, then test it: how does the model's probability estimate compare with the experimental proportion from the sample? This is strand 3 of the standard, and the highest-value fortnight in the course.

Sample: ages of 131 visitors surveyed at a Wellington waterfront café over one week. Drag the parameters and watch the model curve sit over the data.

Test threshold — compare P(X > t)

t = 35

The sentence structure students need

Experimental estimate from the sample: p̂ = … Model estimate from the fitted distribution: P(X > t) = … The model over/under-estimates the proportion of … by about … Therefore this model is / is not suitable for modelling … in this sample.

Excellence needs all four lines. Identifying the over-estimate alone is Merit; the judgement about suitability, in context, is what lifts it.

Module 06 · Where the grade is won

Assumptions trainer

Most students write one sentence and stop at Achieved. The escalation from A to M to E is a fixed, teachable shape. Work through a context, write your own answer, then reveal the three tiers.

The three-tier shape

AName the assumption.

MName it in context, and either give a concrete example of it failing or explain the effect on the estimate.

EName it in context, explain why it may not hold here, and explain the effect on the probability estimate.

Module 07 · Question shapes

The eight archetypes

Every AS91586 paper is built from these. Teach the shapes, not the contexts — the contexts change every year, the shapes never do.

Module 08 · Checkpoint

Diagnostic checkpoint

Ten items targeting the errors that cost marks most often. Feedback names the misconception, not just the answer.