Loading subjects
Probability Concepts · Part 4 of 4
12 exam-style questions with model answers, plus 16 quick multi-choice questions — every question on this part of the standard, grouped by the 4 pages of notes they come from.
Write a full answer before you reveal the model one. That comparison is where the learning happens.
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.