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33 exam-style questions with model answers, plus 44 quick multi-choice questions — every question on the site for this standard, grouped by the 11 pages of notes they come from.
Write a full answer before you reveal the model one — that comparison is where the marks come from. Every block links back to the notes that teach it.
A six-sided die is rolled 300 times and shows a six on 58 occasions. State the model estimate and the experimental estimate of the probability of rolling a six.
A manufacturer's model predicts that 2% of its components are defective. In a sample of 4 000 components, 118 are defective. Compare the two estimates and explain what could account for the difference.
A conservation group models the probability that a monitored kākā nest fledges at least one chick as 0.35, assuming nests fail independently of one another. Over one season, 8 of 40 monitored nests fledged a chick (0.20). The group concludes the model is wrong and revises the probability to 0.20. Discuss.
Define what it means for a process to be random, and explain why a run of six heads in a row does not mean a coin is unfair.
A lottery player notes that the number 17 has not been drawn in the last 40 draws and buys a ticket with 17 on it, saying it is 'overdue'. Explain the error, and explain why the law of large numbers does not support the player's reasoning.
A regional health service notices that a rare childhood illness, which affects about 1 child in 20 000 per year nationally, has affected 3 children in one small town of 4 000 people within 18 months. Residents demand an investigation into a local cause. Discuss what probability theory says about this cluster and how the health service should respond.
For two events, P(A) = 0.5, P(B) = 0.3 and P(A and B) = 0.2. Find P(A or B) and P(neither A nor B).
A tramping club finds that 45% of members own a tent, 60% own a portable stove, and 25% own neither. A member is chosen at random. Find the probability the member owns both, and the probability the member owns exactly one item.
A regional airline finds that on any given flight the probability of a delay is 0.18 due to weather, 0.11 due to a technical issue, and 0.06 due to both together. A traveller has four flights on a trip. She calculates the probability of at least one delayed flight as 1 − (1 − 0.23)4 = 0.648. Evaluate her reasoning and comment on the answer.
Explain the difference between mutually exclusive events and independent events, giving an example of each.
In a town, 30% of households have a heat pump and 25% have solar panels. If these were independent, what proportion would have both? A survey finds 12% have both. Interpret this result.
An insurer models the probability that a given house in a coastal suburb suffers flood damage in a year as 0.02, and treats houses as independent. It insures 3 000 houses in the suburb and calculates that the probability of more than 100 claims in a year is negligible. Discuss the model and its consequences for the insurer.
In a group of 200 people, 90 own a bike and of those 36 also own a car. Find the probability that a randomly chosen person owns both, and the probability that a bike owner owns a car.
A school of 600 students records that 240 study a language, 150 play in the orchestra, and 84 do both. Construct the two-way table and determine whether studying a language and playing in the orchestra are independent.
A hospital compares two surgeons. Surgeon A succeeded in 210 of 250 operations (84%); Surgeon B succeeded in 150 of 200 (75%). Broken down by case severity: for routine cases Surgeon A succeeded in 190 of 210 (90.5%) and Surgeon B in 90 of 95 (94.7%); for high-risk cases Surgeon A succeeded in 20 of 40 (50%) and Surgeon B in 60 of 105 (57.1%). Discuss what these tables show.
In a group of 100 people, 40 own a cat, 55 own a dog and 15 own both. Draw the Venn diagram regions and find how many own neither.
Of 300 café customers, 168 buy coffee, 96 buy cake and 45 buy both. Find the probability a randomly chosen customer buys exactly one item, and the probability a cake buyer also buys coffee. Interpret the second answer.
A gym reports that of its members, 62% use the weights area, 48% use classes and 35% use the pool. It also reports that 30% use weights and classes, 20% use weights and the pool, 22% use classes and the pool, and 12% use all three. Show that these figures are inconsistent, and explain what may have gone wrong.
A bag holds 4 red and 6 blue counters. Two are drawn without replacement. Find the probability both are red.
On any morning the probability of rain in a coastal town is 0.3. If it rains, the probability the harbour ferry is cancelled is 0.45; if it does not rain, the probability of cancellation is 0.08. Find the probability the ferry runs, and the probability that on a randomly chosen morning it rains and the ferry runs.
A factory has three machines producing the same component: machine X makes 50% of output with a 2% defect rate, machine Y makes 30% with a 4% defect rate, and machine Z makes 20% with a 7% defect rate. A quality manager proposes testing every component from machine Z only, arguing it has the highest defect rate. Evaluate this proposal.
P(A and B) = 0.18 and P(B) = 0.45. Find P(A | B) and state what it means.
A vineyard finds that 20% of its grapes come from the older block. Of grapes from the older block, 15% are rejected at sorting; of grapes from the newer block, 6% are rejected. Find the overall rejection rate, and the probability a randomly chosen rejected grape came from the older block.
A university reports that its overall pass rate for first-year papers is 82%. It also reports that the pass rate is 90% for students who attend more than 80% of lectures, and 61% for students who attend less. The Dean concludes: 'Attending lectures raises the pass rate by 29 percentage points, so attendance should be made compulsory.' Discuss.
Explain, with an example, why P(A | B) and P(B | A) are not the same.
A workplace drug test has a 98% sensitivity and a 97% specificity. In a workforce where 2% of employees use the substance, an employee tests positive. Find the probability they are a user, and interpret the result.
In a court case, an expert testifies: 'The DNA sample matches the defendant. The probability of such a match occurring by chance in an unrelated person is 1 in 100 000. Therefore the probability that the defendant is innocent is 1 in 100 000.' The city has a population of 800 000 adults. Evaluate the expert's reasoning.
In a study, 60 of 1 500 exposed people and 40 of 3 000 unexposed people developed a condition. Find the absolute risk in each group and the relative risk.
A health campaign reports that a new preventive measure 'cuts the risk of the illness by 40%'. In the trial, the illness occurred in 25 of 5 000 people using the measure and 42 of 5 040 not using it. Evaluate the campaign's claim.
Two newspapers report the same study of a medication. Paper A: 'Medication triples the risk of a rare blood clot.' Paper B: 'Medication prevents 1 in 40 heart attacks.' The study found clots in 3 per 100 000 on the medication against 1 per 100 000 not on it, and heart attacks in 30 per 1 000 on the medication against 55 per 1 000 not on it. Discuss how a patient should weigh these.
A distribution table for X gives P(0) = 0.2, P(1) = 0.3, P(2) = k and P(3) = 0.15. Find k and then find P(X ≥ 2).
Two fair four-sided dice are rolled and X is the larger of the two numbers showing (or that number if both are equal). Construct the probability distribution of X and describe its shape.
A rural fire service models the number of callouts per day in summer with the distribution P(0) = 0.55, P(1) = 0.28, P(2) = 0.11, P(3) = 0.04, P(4 or more) = 0.02. Over 90 summer days it records: 0 callouts on 42 days, 1 on 22, 2 on 12, 3 on 8, 4 or more on 6. Assess the model.