NCEA 91267: probability methods (Level 2 Maths)

What NCEA 91267 assesses

NZQA says you are assessed on applying probability methods in solving problems, set in real-life or statistical contexts. The standard (version 3) lists the methods:

• risk and relative risk
• the normal distribution
• experimental distributions
• relative frequencies
• two-way tables
• probability trees

The three grades read the same way in the standard, with a different kind of thinking. Achieved applies probability methods in solving problems. Merit uses relational thinking: selecting and carrying out a logical sequence of steps, connecting concepts or representations, and relating findings to a context. Excellence uses extended abstract thinking: devising a strategy, developing a chain of logical reasoning or proof, or forming a generalisation.

The NCEA 91267 exam in 2026 and 2027

NCEA 91267 is one of three Level 2 Mathematics and Statistics externals, with 91261 (algebra) and 91262 (calculus). The 2026 exam is on Thursday 19 November, in the morning session. NZQA's specifications for both years say the exam is on printed paper and 91267 is a 4-credit external.

• Each exam has three questions made up of multiple parts, some of them resource-based and some knowledge-based. Parts are not ordered by difficulty, so attempt every part.
• Calculator: you must bring an approved calculator.
• Where probabilities come from: you may calculate them from one or more tables, written information or a probability tree, and you may need to create a tree from written information.
• Three decimal places: NZQA says all working should be given to a minimum of three decimal places throughout a question.
• Describe and compare: you may be asked to compare distributions from statistics or graphs, with reference to shape, centre and spread and numerical justification, and to calculate and interpret relative risks above and below 1.

How NCEA 91267 is marked

Each of three questions is marked from N0 to E8 and the scores add to a total out of 24. NZQA's report says candidates who did well on one or two of the three components, but little on the third, tended to land at Achieved, so balance matters more here than in the other maths standards.

NCEA 91267 cut scores

Probability has the widest spread of boundaries: Excellence started at 19 in 2024 but 20 in other years, and Merit as low as 13. Each score is where the grade begins, out of 24.

Cut scores 2025

2025 schedule: Achieved from 8, Merit from 14, Excellence from 20. Compared with 2024, Merit went up by 1 and Excellence went up by 1.

Cut scores 2024

2024 schedule: Achieved from 8, Merit from 13, Excellence from 19. Compared with 2023, Merit came down by 1 and Excellence came down by 1.

Cut scores 2023

2023 schedule: Achieved from 8, Merit from 14, Excellence from 20. Compared with 2022, Achieved came down by 1.

Cut scores 2022

2022 schedule: Achieved from 9, Merit from 14, Excellence from 20. Compared with 2021, Achieved went up by 1 and Merit came down by 1.

Cut scores 2021

2021 schedule: Achieved from 8, Merit from 15, Excellence from 20.

Worked examples at each grade

Achieved: a probability tree

The bus is on time 70% of the time. When it is on time, there is a 10% chance of being late to school; when it is late, a 60% chance. Find the probability of being late to school. Multiply along each branch, then add the branches that end in "late":

Merit: conditional probability and relative risk

Given that you were late to school, what is the probability that the bus was late? Divide the late-bus branch by the total:

Now relative risk. In a study of 400 teenagers, 150 slept for less than 7 hours a night and 60 of them got a cold; of the other 250, 50 got a cold. The risk is 60 ÷ 150 = 0.4 for the short sleepers and 50 ÷ 250 = 0.2 for the rest:

Merit asks you to interpret it: the short sleepers were twice as likely to catch a cold.

Excellence: the normal distribution, and a justified claim

Journey times to school are normally distributed with a mean of 17 minutes. 80% of journeys take less than 20 minutes. Find the standard deviation. The z-score for a cumulative probability of 0.80 is 0.8416, so:

Excellence goes further: compare two distributions using shape, centre and spread with numbers, and say whether a claim such as "short sleepers get twice as many colds" is valid. A relative risk of 2 shows an association in this sample, but it does not show that short sleep caused the colds, and 400 teenagers may not represent all teenagers.

Now try some normal distribution questions 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

The Level 2 Maths track shown above is a short timed mock of 8 questions, not a full exam and not written by NZQA (NCEA Level 2 Maths). It practises algebra and functions, not probability. For probability, use the NCEA Level 3 Statistics track, whose normal distribution, conditional probability and probability and Venn diagram questions overlap with 91267, and treat its hypothesis testing, sampling and regression questions as extension.

Is NCEA 91267 changing?

The 2027 specification for Level 2 maths is unchanged for probability. After 2027 the Year 12 qualification becomes the NZ Certificate of Education in 2029.

Common mistakes in NCEA 91267

• Adding when you should multiply. NZQA's 2025 report lists mixing up when to add and when to multiply as a common Not Achieved error: multiply along a branch, add across branches.
• Probabilities outside 0 to 1. A probability can never be more than 1 or less than 0.
• Stopping at the z-value. The report says some candidates found z but could not use it for inverse normal problems or to find the standard deviation.
• Rounding too early. The report says probabilities should be calculated to at least four decimal places.
• Calculating without interpreting. Higher grades need relative risk and distributions interpreted in context, with the correct statistical vocabulary.

Read next

New Zealand · NCEA Level 2NCEA Level 2 Maths: standards, exam date and practiceNCEA Level 2 Maths explained: the algebra, calculus and probability externals, the internals, the 2026 exam date, 2027 dates and worked examples for each.