NCEA 91585: apply probability concepts in solving problems (Level 3 Statistics)

What NCEA 91585 assesses

NZQA's standard says you apply probability concepts to problems set in real-life or statistical contexts. The methods come from this list:

• true probability versus model estimates versus experimental estimates
• randomness, independence and mutually exclusive events
• conditional probabilities
• probability distribution tables and graphs
• two-way tables, probability trees and Venn diagrams

Achieved selects and uses methods, shows knowledge of concepts and terms, and communicates with suitable representations. Merit uses relational thinking: a logical sequence of steps, connecting representations, and relating findings to the context. Excellence uses extended abstract thinking: devising a strategy, developing a chain of reasoning, making a generalisation, and using contextual knowledge to reflect on the answer.

The NCEA 91585 exam in 2026 and 2027

NCEA 91585 is one of three Level 3 Statistics externals, with 91584 (evaluating statistically based reports) and 91586 (probability distributions). The 2026 Level 3 Statistics exam is on Tuesday 24 November, in the morning session. NZQA's 2027 timetable is not out yet: it says exams run from 9 November to 3 December 2027. The 2026 and 2027 specifications match: printed paper, a Formulae and Tables Booklet provided, and an approved calculator, preferably a graphing calculator.

• Show your method. NZQA's specification says you should clearly show the method you used to calculate probabilities and state your assumptions.
• Sources of probabilities. Expect them from formulae, a distribution table or graph, tables of counts, simulation results, or written information.
• Round late. Early rounding may be penalised.

How NCEA 91585 is marked: grade score marking

Three questions are each scored N0 to E8 and added. Here a probability tree or table with stated assumptions earns u units, linking a result to the context earns r, and judging a model against experimental results earns t.

NCEA 91585 cut scores

These are NZQA's cut scores for the total out of 24, taken from each year's assessment schedule. Over these five years Achieved needed 7 or 8, Merit 13 to 15 and Excellence always 19. The 2026 cut scores will appear in NZQA's 2026 schedule once the exam has been marked.

Worked examples at each grade

Suppose 200 Year 13 students are surveyed: 80 play a sport, 60 have a part-time job, and 25 do both.

Achieved: conditional probability from a Venn diagram

Find the probability that a student has a job, given that they play a sport. Restrict attention to the 80 who play a sport. Of those, 25 have a job:

Merit: test independence and link to the context

Are playing a sport and having a job independent? Compare P(job) = 60 ÷ 200 = 0.30 with P(job | sport) = 0.3125. They are close, and P(sport) × P(job) = 0.4 × 0.3 = 0.12 is close to P(both) = 25 ÷ 200 = 0.125. The probability that a student does at least one is 0.4 + 0.3 − 0.125 = 0.575. Merit needs the sentence that follows: in this sample the two are close to independent, so having a job makes a student only slightly more likely to play a sport.

Excellence: model, true and experimental probabilities

A bus is late with probability 0.2 on any day, independently. The model gives the probability that none of 5 school-day buses is late as:

A student simulates 10 five-day weeks and sees no late bus in only 1 week, an experimental estimate of 0.1. Excellence evidence explains that the model predicts about 3.3 such weeks in 10, and that an experimental estimate from 10 trials varies a lot (its standard error is about 0.15), so 1 week is not strong evidence against the model. More trials would give an estimate closer to the model, which itself rests on the assumptions of a fixed 0.2 and independence from day to day.

Now try an independence question in our Probability course, then press Mark to check your answer in the course:

Exam answers

The answers and a worked mark scheme come with today's paper once it is marked. We do not publish official or past paper answers. Every question on the daily paper is our own.

Use the daily paper as a practice exam

Uni-Sine publishes a free daily paper on the NCEA Level 3 Statistics track. It is a short timed paper of 8 questions worth 25 to 35 marks, not a full NCEA exam, and it is not written by NZQA. Its topics include probability and Venn diagrams and conditional probability, which are the core of 91585. For the whole subject, including credits and endorsements, see the NCEA Level 3 Statistics guide.

Common mistakes in NCEA 91585

• Answers with no method. NZQA's 2025 report says candidates must read carefully, write answers in context, and support them with calculations.
• Blank parts. The 2025 report says a significant number of candidates leave parts blank. Restart at part (b) if part (a) stalls.
• Rounding early. Carry full values through, and round the final answer.
• Generalising too far. A sample from one school or one season tells you about that group, so say what you cannot conclude.

Read next

New Zealand · NCEA Level 3NCEA Level 3 Statistics: standards, exam date and practiceNCEA Level 3 Statistics explained: the internals, the three externals on reports, probability and distributions, the 2026 exam date and worked examples.