3. (a) A bag contains 3 red pills, 5 white pills and 7 blue pills. If 5 pills are selected at
random, what is the probability that the 5 selected pills contain
(i) exactly 1 red pill and 3 blue pills?
(ii) either exactly 1 red pill, or 3 blue pills?
[3 marks]
[5 marks]
(b) Suppose that a particular type of cancer affects 1% of the population. A new test
has been invented for detecting this particular type of cancer but it's not perfect:
although the test gives a positive result for 85% of people who have the cancer,
it also gives a positive result for 5% of the people who are cancer-free. Suppose
that the test is now applied to a random person about whom we have no relevant
information relating to this particular type of cancer (apart from the fact that
he/she comes from this population). Let A be the event that the test result is
positive and let B be the event that a person has this particular type of cancer.
Calculate the probabilities for following events:
(i) that the test result will be positive;
[3 marks]
(ii) that, given a positive result, the person has this particular type of cancer
(please round your final answer to 3 decimal places);
[3 marks]
(iii) that, given a negative result, the person is cancer-free (please round your final
answer to 3 decimal places).
[3 marks]
(iv) Furthermore, are events A and B independent?
[3 marks]