Apply probability concepts

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 questions every year, each opening with a Venn/table/graph read-off, then building to a claim you must test with a correctly-oriented conditional probability, then closing on a reliability judgement (which estimate to trust, whether a die is biased, whether an assumption holds). Grades below are read from each year's own evidence statement.

Across 4 years

The pattern — what to prepare

The 2022 paper ↗ Marking schedule ↗

Question One — Technology and language subject choices

167 Year 10 students chose option subjects: 62 chose technology, 138 did not choose a language, and 10 chose both technology and a language.

• n = 167
• Technology: 62; not-language: 138; technology & language: 10

• a What proportion of students had not chosen a technology subject, but were studying a language? Achieved
• b Given all three of three randomly chosen students were studying a technology subject, find the probability that all three were NOT studying a language. Merit
• c The school claims choosing a technology subject has no effect on the probability of also choosing a language. Investigate the validity of this claim using statistical reasoning. Excellence
• d(i) 475 senior students: 71.6% intending university, 63.4% intending university AND following their preferred pathway, 25.3% not intending university but following their preferred pathway. Calculate the number of senior students who were not intending on going to university, and not intending on following their preferred pathway. Merit
• d(ii) It is claimed students are twice as likely to follow their preferred pathway if intending on university than if not. Comment on whether this data supports the claim. Excellence
• e Of NZ Level 3 Statistics students: 12% entered all three exams, 14% only 91585, 10% only 91586, 57% entered for 91585 and 91586, 17% for 91584 and 91586. Given a student was entered for 91586, calculate the number also entered for 91584. Merit

Question Two — University lecture attendance and enrolment

Auckland University male/female enrolment 2017-2020; Canterbury University first-year students, with attendance rates differing by whether their courses had final exams.

• Male students: 18,168 (2017) to 18,065 (2020); female 24,134 to 24,760
• Of 50 first-years, 9 took courses with no final exam; 38% of exam-course students attended lectures regularly vs 21% of no-exam-course students

• a(i) Which year (2017-2020) had the greatest proportion of male students at Auckland University? Support with calculations. Achieved
• a(ii) Give TWO reasons why the calculated proportions are only estimates of the true probability of a student being male that year. Merit
• b(i) Calculate the probability that a randomly selected Canterbury first-year student attended lectures regularly. Achieved
• b(ii) A student believed P(E∩A) + P(E′∩A′) = 1. Explain, using statistical reasoning, whether this student was correct. Merit
• b(iii) Calculate the probability that three of four randomly chosen students took a course without a final exam AND attended lectures regularly, explaining any assumptions made, and comment on one issue applying this result to all students in 2022. Excellence

Question Three — Are digits on NZ number plates equally likely?

A statistics class investigated whether the numerical digits on standard NZ number plates occur equally often, comparing individual students' small samples against a simulation of the whole class's pooled data (n = 2764).

• a(i) With no prior knowledge, what theoretical probability would you give to each digit appearing? Justify your answer. Achieved
• a(ii) From a given bar chart, calculate the probability a digit was 5 or higher, given it was less than 8. Achieved
• b(i) A student claims two other students must have deliberately looked for number plates with the digit 8. Evaluate this claim using statistical reasoning and calculations. Merit
• b(ii) Explain why it is not possible to conclude from a few students' small bar charts that digits are not equally likely. Merit
• c Using the simulation of the whole class's pooled data (n = 2764) and the ideas of true, model, and experimental probability, what conclusions can be drawn about whether digits are equally likely? Excellence

The 2025 paper ↗ Marking schedule ↗

Question One — Badminton, table tennis, hockey

The 2023 School Sports Census surveyed 144,862 students across 91 sports.

• Badminton: 11,195; table tennis: 2,125; hockey: 13,304
• Badminton & table tennis: 800; table tennis & hockey: 600; badminton & hockey: 1,100; all three: 500

• a Estimate the probability that a randomly selected student participates in only one of these three sports. Achieved
• c Interpret the relationship between "student plays badminton" and "student plays hockey" — explain whether these events are independent, using calculations and statistical reasoning. Excellence
• d(ii) A school claims more football injuries occur on artificial turf (80% of games, 0.04 injury probability) than on natural grass (0.10 injury probability). Discuss why care should be taken using this data to make that claim. Merit
• d(iii) Estimate the probability that, from three randomly selected football players, only one is injured during their game — stating and justifying any assumption(s) made. Excellence

Question Two — A sideline concussion test

A study of 298 rugby players with head knocks compared a sideline test result against a later medical assessment.

• P(positive test & concussion) = 0.074
• P(positive test & no concussion) = 0.124
• P(negative test & concussion) = 0.131
• P(negative test & no concussion) = 0.671

• c(ii) A safety guideline states "most head knocks in school rugby do not result in concussion." Does the data from this study support this claim? Merit
• c(iii) A coach says: "If a player gets a positive sideline test result, there's a 'one in four' chance they actually have a concussion." Does the data support this claim? Support your answer with statistical reasoning. Excellence
• d Compare the true probability, model estimate, and experimental estimate for cricket participation in 2023, and discuss which is most reliable for predicting future participation.
  • 2023 census: 8,880 of 50,000 played cricket
  • Random sample: 120 of 1,000 played cricket
  • Coordinators' estimate (model): 25%
  Excellence

Question Three — Ultimate frisbee, and a netball tryout

Census data on ultimate frisbee and archery participants, 2005–2020; a netball tryout observer classifies players as selected or not, against a coach's 90% accuracy goal.

• At-standard players: 42 of 50 correctly selected
• Below-standard players: 12 of 25 incorrectly selected

• a(ii) Explain why the probabilities calculated from the ultimate frisbee/archery graph might not correctly represent the true probability of a student playing ultimate frisbee, compared to other sports, in those years. Merit
• c(ii) Comment on the effectiveness of the observer in selecting the correct players for the squad, supporting your answer with at least one calculation, against the coach's goal of 90% accuracy. Excellence

The 2024 paper ↗ Marking schedule ↗

Question One — An AI writing detector, and exam cheating

A writing detector has a 1% false positive rate and misses AI-generated writing 74% of the time; 22% of a class's writing is estimated to be AI-generated.

• False positive rate: 1%
• Miss rate: 74%
• Estimated AI-generated writing: 22%

• a(i) If 120 pieces of writing are screened by this detector, approximately how many would be detected as potentially AI-generated? Achieved
• a(ii) A piece of writing is detected as potentially AI-generated. Comment on whether a teacher should be concerned that the student could be unfairly accused of cheating, supporting your answer with statistical reasoning. Merit
• b(i) Comment on whether "student cheats" and "student completes the assessment in less than 25% of the allocated time" are independent, using statistical reasoning, and interpret the relationship between the two events.
  • 1% of students cheat
  • 20% complete the assessment in under 25% of allocated time
  • 80% of cheaters complete it in under 25% of allocated time
  Excellence

Question Two — Data-science skill sets, and NCEA Level 2 attainment

35 job applicants were assessed for three skill sets (coding, maths/stats, subject-specific understanding).

• 7 applicants had none of the three skills
• 9 applicants had exactly two of the three skills

• a(ii) Calculate the proportion of applicants who had only one of the required skills. Merit
• a(iii) It is claimed that applicants with maths/stats knowledge are twice as likely to also have subject-specific understanding as to have coding skills. Does this data support the claim? Excellence
• b(i) Based on the data, is the suggestion that the total proportion of 18-year-olds with NCEA Level 2 or above has increased between 2012 and 2022 correct? Support your answer with statistical reasoning.
  • NCEA L2+ attainment: 77.3% (2012) → 82.4% (2022), peaking at 85.1% in 2018
  Merit

Question Three — Dice games, and a possibly-biased die

A player plays two dice games (Game A: win if the total is 5–10 inclusive; Game B "Hazard": a point-based win/loss sequence), then suspects one of their dice is biased.

• 1,000 rolls of the suspect die: 1→138, 2→189, 3→197, 4→143, 5→179, 6→154

• a(i) For Game A, calculate the probability that a player wins at least once when playing 3 times. Achieved
• b(i) For these 1,000 rolls, which is more likely: rolling 3 or less, or rolling 4 or more? Support your answer with statistical calculations. Achieved
• b(ii) Should the player be concerned that one of their dice is biased? Use the simulation results and the observed rolls, referring to experimental, theoretical, and true probability as part of your answer. Excellence

The 2023 paper ↗ Marking schedule ↗

Question One — Coffee preference by year group

Combined survey data from Year 9, 11, and 13 students on liking coffee.

• Year 9: 13 like / 43 dislike
• Year 11: 11 like / 38 dislike
• Year 13: 22 like / 30 dislike (n = 157)

• a(i) Calculate the probability of a randomly selected student being in Year 9 or Year 11, given that they stated they do not like coffee. Achieved
• a(ii) Four Year 11 students are randomly chosen. Calculate the probability that at least one of these four stated they like coffee. Merit
• a(iii) State an assumption you made when calculating (ii), and discuss whether this assumption is likely to be valid. Excellence

Question Two — Coffee-type preference, and an eikosogram

510 people who liked coffee were asked which of mocha, flat white, or cappuccino they preferred (up to two selections allowed).

• 423 preferred at least one type
• 81 cappuccino only; 38 flat white only; 29 flat white & mocha; 103 mocha only; 0 cappuccino & mocha

• a(i) Calculate the proportion of people who indicated they preferred cappuccino. Achieved
• a(ii) A claim states it is 50% more likely that people who preferred cappuccino selected ONLY cappuccino, compared to people who preferred mocha selecting ONLY mocha. Do the survey results support this claim? Excellence
• b(ii) An eikosogram splits the same 510 people by rural/urban home area and cafe/home coffee preference. What evidence from the eikosogram shows a difference in coffee-location preference between rural and urban people? Support your answer with statistical reasoning, with reference to sampling variation.
  • Rural: 28.0% of sample, P(cafe | rural) ≈ 0.175
  • Urban: 72.0% of sample, P(cafe | urban) ≈ 0.515
  Excellence

Question Three — Coffee brands

Three friends compared the strength and Fair Trade status of 15 coffee brands they had tried.

• 15 brands tried, 12 Fair Trade
• 8 extra strong, of which 5 were Fair Trade

• a Explain whether "the coffee is not a Fair Trade brand" and "the coffee is not extra strong" are mutually exclusive, giving at least one numerical calculation. Merit

Every claim-evaluation part rewards a stated verdict (supported / not supported / partly supported) with the calculated probability as the reason — a correct number with no verdict, or a verdict with no number, 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.