Apply probability distributions

Practice

Recall it first, then use it — testing yourself builds the memory, where rereading the notes above only feels easier.

Recall and application shuffled together — you have to work out what each item wants before you can answer it, which is the part the exam actually tests. Both kinds feed the same spacing schedule, so anything you miss comes back sooner.

Say the answer out loud before flipping — that attempt is where the learning happens, even when you get it wrong.

Exam-style only. Instant feedback, shuffled each attempt, best score saved on this device.

What actually comes up

Three real scenarios every year, mixing binomial, Poisson, normal, and triangular/uniform models in the same paper. Every calculation part expects the distribution NAME and its PARAMETERS stated explicitly, and every model is eventually checked against real data (a histogram, a simulation, a stated proportion) and judged for fit — a calculation with no fit-judgement is only half the question.

Across 4 years

The pattern — what to prepare

The 2025 paper ↗ Marking schedule ↗

Question One — Post lengths on X

Post lengths for one group of X (Twitter) users follow a normal distribution; a short post is fewer than 80 characters.

• μ = 150 characters, σ = 30 characters
• Free-account character limit: 280

• a(i) Use this normal distribution to estimate the probability that a post is NOT short (fewer than 80 characters). Achieved
• a(iii) State an assumption you made in the previous calculation, and describe how violating it could affect the accuracy of your estimate. Merit
• a(iv) Without further probability calculations, use statistical reasoning to explain why the 280-character limit is unlikely to affect this group of users. Excellence

Question Two — Social media user ages

A sample of 131 adult social media users aged 17–82 has its age distribution shown in a histogram, with the largest frequency between 25 and 30 years old.

• n = 131, ages 17–82
• Triangular model: a = 17, b = 82, c = 30

• b(i) Justify the choice of the triangular distribution parameters a = 17, b = 82, c = 30, including an observation from the histogram. Merit
• b(ii) Use this triangular distribution model to estimate the probability that a randomly chosen user is more than 35 years old. Merit
• b(iii) Use your answer from (ii) and observations from the histogram to comment on the suitability of the triangular model for this sample. Excellence

Question Three — Screening for harmful content

An automated system screens posts for harmful content in a population where 1 in 10 posts are actually harmful, and separately reports false positives on harmless content.

• P(harmful) = 0.1
• False positives per batch of 100: mean 2.1

• a Use a probability distribution model to estimate the probability of observing fewer than 3 harmful posts among 20 screened, stating the distribution name and parameters. Achieved
• b Describe the independence assumption used in (a) in context, and discuss whether it is valid for this situation. Merit
• d A competing system reports at least one false positive in 97% of its batches of 100. Calculate its false-positive rate and use it to compare the accuracy of the two systems. Excellence

The 2024 paper ↗ Marking schedule ↗

Question One — Innies, outies, and fluff

The world's first belly-button survey found 96% of respondents had an "innie" belly button; a separate survey found 66% had fluff in their belly button.

• 96% innie (4% outie)
• 66% have belly-button fluff

• a(i) Estimate the probability that in a class of 30 students, no student has an outie belly button, stating the distribution model and parameters. Achieved
• a(ii) Give TWO reasons why the distribution chosen in (i) is appropriate. Merit
• a(iii) Estimate the probability that one student in a class of 30 has an outie belly button AND it contains fluff. Merit

Question Two — The Nature of Navel Fluff study

A 2009 study (Steinhauser) recorded the weight of the author's own daily belly-button fluff over 502 days; a triangular distribution was proposed to model it.

• n = 502 days
• Triangular model: min = 0, max = 7.4, mode = 1.0 mg

• a(ii) Using the given parameters, estimate the probability that on any given day the fluff weight is less than 1.5 mg. Achieved
• a(iii) Suggest alternative parameters for a triangular distribution that would fit the data better, and justify your choice. Excellence
• b(iii) T = towels used per person per day; C = clothing changes per day. SD(C) = 0.5; SD(T + C) = 1.754. Are T and C independent? Comment on what this suggests, supporting your answer with statistical statements and calculations. Excellence

Question Three — Belly-button washing frequency

A 2012 study of 138 people recorded belly-button washing frequency per 7-day week; a media headline claimed "less than a quarter of US citizens wash their belly button daily."

• n = 138
• Poisson model: λ = 2.8 washes/week

• a Comment on whether the headline is correct, supporting your answer with statistical calculations and reasoning. Merit
• b(i) Use a Poisson model (λ = 2.8/week) to estimate the probability that a randomly selected person washes their belly button at least once in a day. Achieved
• b(iv) Based on a 1000-trial simulation compared with the observed data, discuss whether the Poisson model appears appropriate for this data. Excellence

The 2023 paper ↗ Marking schedule ↗

Question One — Sheep stocking rates, and wool fineness

Average sheep stocking rates on NZ hills and mountains are 9 sheep per hectare; sheep are known to flock together. A separate sample of 49 Awassi sheep has its wool fineness (in microns) shown in a histogram.

• Average rate: 9 sheep/hectare
• Wool fineness sample: n = 49

• a(i) Use a suitable probability distribution to estimate the probability of more than 10 sheep on a given hectare, stating the distribution name and parameters. Achieved
• a(ii) Give TWO reasons why sheep flocking together may affect your choice of distribution and its required conditions. Merit
• b(i) Using a uniform distribution, calculate the probability that a randomly selected Awassi sheep's wool fineness is greater than 35 microns, stating the parameters used. Achieved

Question Two — A hen's egg-laying rate

A hen's egg-laying rate is modelled by a Poisson distribution; she lays at least one egg each week 99.73% of the time.

• a(i) Calculate an estimate for the average number of eggs the hen lays each week. Achieved
• a(ii) What is the most likely number of eggs the hen will lay each week? Justify your answer with statistical reasoning or calculations. Merit
• a(iii) Each of the 10 eggs independently does or doesn't hatch, with probability 0.65. Calculate the probability that 9 or more of the 10 eggs hatch.
  • Hatch probability per egg: 0.65
  • 10 eggs laid, each independent
  Excellence

Question Three — Dairy cow milk production

Total milk production per NZ dairy cow per season is modelled as normal; for South Island cows specifically, approximately 30% produce less than 4290 kg per season.

• NZ model: μ = 4370 kg, σ = 1350 kg
• South Island: 30% produce < 4290 kg

• a(i) Discuss whether the normal distribution model described is appropriate for modelling South Island dairy cow milk production, supporting your answer with statistical calculations. Excellence

The 2022 paper ↗ Marking schedule ↗

Question One — Retail purchases

For a NZ clothing retail company, a purchase is made by 35% of customers entering their stores. The company has many stores throughout New Zealand.

• p = 0.35 per customer
• n = 10 customers per store

• a(i) Using an appropriate probability distribution model, calculate an estimate for the probability that fewer than 3 of the next 10 shoppers make a purchase. Achieved
• a(ii) Give TWO assumptions made for the probability distribution model used in (i). Merit
• a(iii) Using an appropriate probability distribution model, calculate an estimate for the probability that at least 3 out of the next 10 customers make a purchase in each of 4 different stores. Merit
• a(iv) An assumption was made when calculating the probability estimate in (iii). Justify this assumption in context. Excellence
• b A "click and collect" order is ready any time between 24 and 144 hours (uniform). A "home delivery" order time is normal with mean 96 hours, standard deviation 38.4 hours. Two friends each order at the same time, one by each method. Using appropriate probability model(s), estimate the probability that both friends wait more than 120 hours, justifying any assumptions made. Excellence

Question Two — Items purchased and defect rates

The number of items purchased per 20-minute interval is modelled by a Poisson distribution with λ = 2.8. Stock arrives in boxes of 50, checked for defects; defect counts are given for two suppliers.

• λ = 2.8 items per 20 min
• Supplier A: P(D=0..3) = 0.73, 0.16, 0.09, 0.02
• Supplier B: P(D=0..3) = 0.66, 0.23, 0.07, 0.04

• a(i) Use the Poisson model to estimate the probability that a customer purchases fewer than 4 items in a 20-minute interval. Achieved
• a(ii) Give TWO reasons why the Poisson distribution may not be appropriate to model this situation. Merit
• a(iii) What value would λ need to be so that at least 99% of customers purchase at least 1 item in a 20-minute interval? Excellence
• b(i) Calculate the mean and standard deviation of the number of defects per box of 50 items from Supplier A. Achieved
• b(ii) Using statistical reasoning, explain which supplier better fulfils the requirement of low, consistent defect counts. Merit

Question Three — Customer spend per transaction

A retail company records spend per transaction over 100 transactions: mean

• n = 100 transactions
• μ =
  27.75, σ = $46.22
• mode =
  30, max = $250

• a It is claimed that over 25% of transactions are more than
  50. Use a calculation to decide whether this claim is correct. Achieved
• b Use a normal distribution model to estimate the probability that a randomly chosen customer spends more than
  50. Achieved
• c Suggest an alternative model for customer spend at this store, stating its parameters. Merit
• d Use your alternative model to estimate the probability that a customer spends more than
  50, given that they spend more than
  30. Merit
• e Compare your two probability estimates with the observed proportion from the data. Which model gives the closer estimate? Excellence
• f Recommend which model should be used for this store's spend data, justifying your choice by referring to the features of each distribution. Excellence

A model-fit judgement (over/underestimate, appropriate/not appropriate) rewards a stated verdict with the calculation as the reason — a correct probability with no verdict on fit, or a verdict with no calculation, both fall short of the top band.

Past exams & resources

⏱ Timed opens the paper and starts a countdown matching the exam length. Official NZQA PDFs. If a link 404s, that year’s paper may sit under a slightly different code: browse the NZQA subject page.