Question

Suppose that 44,000 people in New Zealand (population 4,400,000) carry a particular rare gene X that places them at a higher risk of developing cancer. A medical test will correctly indicate the presence of gene X with 0.9 probability when carried out on a person who carries gene X. If the person does not carry gene X it will incorrectly indicate the presence of gene X with probability 0.05.

(i) Draw a fully labelled tree diagram where the first branch represents whether a randomly selected person in NZ carries gene X, and the second branch represents whether the patient tests positive for carrying gene X. Use your tree diagram to find the following:

(ii) (a) the probability that the randomly selected person tests positive for gene X; (b) without using the word probability, write a sentence to explain your answer from part (a) in context.

(iii) (a) the probability that a person who tests positive is actually carrying gene X. (b) without using the word probability, write a sentence to explain your answer from part (a) in context.

(iv) Suppose the test is positive, and a second test is then carried out on the person (suppose this second test is independent of the first). If this second test is also positive, what is the probability the person is carrying gene X? [Hint: you may wish to extend the tree diagram by adding two more branches].

Answer 1