Question 2 [10 Marks]

Three players, A, B and C, take turns tossing a fair six-sided die in the following sequence: ABC ABC ABC ... This continues until one of the players rolls a six. That player then drops out of the game and the others continue until one of them rolls a six, at which point the game ends.

a) [Total of 10 marks for this part] What are the probabilities for each of A, B and C to be the first to exit the game? (Hint: Think carefully about what can happen at each step for A to find the probability A exits first, then consider B and C in turn. The formula for a geometric sum may be useful.)

b) [BONUS: up to 5 marks to be used up to a max of 50] What are the probabilities of the exit sequences AB, BA and CB?

Question 3 [10 Marks]

In a population presenting for testing at a particular neurology clinic, 5% will later develop Huntington's disease. There are two tests used at the clinic. Test 1 identifies 5% of those tested as likely to develop Huntington's disease, whereas test 2 identifies 6% as likely. Both tests are imperfect and give false positive and false negative results.

a) What probabilities (joint, marginal or conditional) are reported in the above text?

b) Given that test 2 produces three times more false positives than test 1 but a third as many false negatives, find the false positive and false negative rates for both tests.

c) The two tests are conditionally independent, which means that