Standard two-outcome diagnostic test

6 It has been estimated that \(90 \%\) of paintings offered for sale at a particular auction house are genuine, and that the other \(10 \%\) are fakes. The auction house has a test to determine whether or not a given painting is genuine. If this test gives a positive result, it suggests that the painting is genuine. A negative result suggests that the painting is a fake. If a painting is genuine, the probability that the test result is positive is 0.95 .
If a painting is a fake, the probability that the test result is positive is 0.2 .

1. Copy and complete the probability tree diagram below, to illustrate the information above.
   
   [diagram]
   Calculate the probabilities of the following events.
2. The test gives a positive result.
3. The test gives a correct result.
4. The painting is genuine, given a positive result.
5. The painting is a fake, given a negative result. A second test is more accurate, but very expensive. The auction house has a policy of only using this second test on those paintings with a negative result on the original test.
6. Using your answers to parts (iv) and (v), explain why the auction house has this policy. The probability that the second test gives a correct result is 0.96 whether the painting is genuine or a fake.
7. Three paintings are independently offered for sale at the auction house. Calculate the probability that all three paintings are genuine, are judged to be fakes in the first test, but are judged to be genuine in the second test.

4 It has been estimated that \(90 \%\) of paintings offered for sale at a particular auction house are genuine, and that the other \(10 \%\) are fakes. The auction house has a test to determine whether or not a given painting is genuine. If this test gives a positive result, it suggests that the painting is genuine. A negative result suggests that the painting is a fake. If a painting is genuine, the probability that the test result is positive is 0.95 .
If a painting is a fake, the probability that the test result is positive is 0.2 .

1. Copy and complete the probability tree diagram below, to illustrate the information above. \includegraphics[max width=\textwidth, alt={}, center]{f3d936ba-8f60-4350-a5b3-92200996434c-3_466_667_834_737} Calculate the probabilities of the following events.
2. The test gives a positive result.
3. The test gives a correct result.
4. The painting is genuine, given a positive result.
5. The painting is a fake, given a negative result. A second test is more accurate, but very expensive. The auction house has a policy of only using this second test on those paintings with a negative result on the original test.
6. Using your answers to parts (iv) and (v), explain why the auction house has this policy. The probability that the second test gives a correct result is 0.96 whether the painting is genuine or a fake.
7. Three paintings are independently offered for sale at the auction house. Calculate the probability that all three paintings are genuine, are judged to be fakes in the first test, but are judged to be genuine in the second test.

2 During an outbreak of a disease, it is known that \(68 \%\) of people do not have the disease. Of people with the disease, \(96 \%\) react positively to a test for diagnosing it, as do \(m \%\) of people who do not have the disease.

1. In the case \(m = 8\), find the probability that a randomly chosen person has the disease, given that the person reacts positively to the test.
2. What value of \(m\) would be required for the answer to part (i) to be 0.95 ?

7 A particular disease occurs in a proportion \(p\) of the population of a town. A diagnostic test has been developed, in which a positive result indicates the presence of the disease. It has a probability 0.98 of giving a true positive result, i.e. of indicating the presence of the disease when it is actually present. The test will give a false positive result with probability 0.08 when the disease is not present. A randomly chosen person is given the test.

1. Find, in terms of \(p\), the probability that
   1. the person has the disease when the result is positive,
   2. the test will lead to a wrong conclusion. It is decided that if the result of the test on someone is positive, that person is tested again. The result of the second test is independent of the result of the first test.
   3. Find the probability that the person has the disease when the result of the second test is positive.
   4. The town has 24000 children and plans to test all of them at a cost of \(\pounds 5\) per test. Assuming that \(p = 0.001\), calculate the expected total cost of carrying out these tests.

1. A disease is known to be present in \(2 \%\) of a population. A test is developed to help determine whether or not someone has the disease.

Given that a person has the disease, the test is positive with probability 0.95
Given that a person does not have the disease, the test is positive with probability 0.03

1. Draw a tree diagram to represent this information. A person is selected at random from the population and tested for this disease.
2. Find the probability that the test is positive. A doctor randomly selects a person from the population and tests him for the disease. Given that the test is positive,
3. find the probability that he does not have the disease.
4. Comment on the usefulness of this test.

Of the cars that are taken to a certain garage for an M.O.T. test, 87% pass. However, 2% of these have faults for which they should have been failed. 5% of the cars which fail are in fact roadworthy and should have passed. Using a tree diagram, or otherwise, calculate the probabilities that a car chosen at random

1. should have passed the test, regardless of whether it actually did or not, [4 marks]
2. failed the test, given that it should have passed. [3 marks]

The garage is told to improve its procedures. When it is inspected again a year later, it is found that the pass rate is still 87% overall and 2% of the cars passed have faults as before, but now 0.3% of the cars which should have passed are failed and \(x\)% of the cars which are failed should have passed.

1. Find the value of \(x\). [8 marks]

In this question you must show detailed reasoning. A disease that affects trees shows no visible evidence for the first few years after the tree is infected. A test has been developed to determine whether a particular tree has the disease. A positive result to the test suggests that the tree has the disease. However, the test is not 100% reliable, and a researcher uses the following model. • If the tree has the disease, the probability of a positive result is 0.95. • If the tree does not have the disease, the probability of a positive result is 0.1.

1. It is known that in a certain county, \(A\), 35% of the trees have the disease. A tree in county \(A\) is chosen at random and is tested. Given that the result is positive, determine the probability that this tree has the disease. [3]

A forestry company wants to determine what proportion of trees in another county, \(B\), have the disease. They choose a large random sample of trees in county \(B\). Each tree in the sample is tested and it is found that the result is positive for 43% of these trees.

1. By carrying out a calculation, determine an estimate of the proportion of trees in county \(B\) that have the disease. [4]