Problem 1 [15 marks]

Three types of infections in a population, which we call A, B and C for simplicity, are mutually exclusive: they cannot be present together. Infection A afflicts one per cent of the population, infection B afflicts one-half of one per cent of the population, and infection C afflicts two per cent of the population.

Sometimes, one of these infections can be present without the victim having any symptoms. Suppose that 10% of the A-infected people have no symptoms, 5% of B-infected people have no symptoms, and 25% of C-infected people have no symptoms. Symptoms are present only when an infection is present, and the symptoms are identical for all three types of infections.

(a) If a person has symptoms, what are the probabilities that he/she has infections A? B? C? [5 marks]

(b) If a person has no symptoms, what are the probabilities that he/she has infections A? B? C?

[5 marks]

(c) A test with a sensitivity of 95% and a specificity of 90% has been developed for infection A. The test is only applied to people who report symptoms.

(i) What is the probability that a person testing negative doesn't have infection A? [2.5 marks]

(ii) What is the probability that there will be a screening error based on this test? [2.5 marks]

Problem 1 [15 marks Three types of infections in a population, which we call A, B and C for simplicity, are mutually exclusive: they cannot be present together. Infection A afflicts one per cent of the population, infection B afflicts one-half of one per cent of the population, and infection C afflicts two per cent of the population. Sometimes, one of these infections can be present without the victim having any symptoms. Suppose that 10% of the A-infected people have no symptoms, 5% of B-infected people have no symptoms, and 25% of C-infected people have no symptoms. Symptoms are present only when an infection is present, and the symptoms are identical for all three types of infections. (a) If a person has symptoms, what are the probabilities that he/she has infections A? B? C? [5 marks] (b) If a person has no symptoms, what are the probabilities that he/she has infections A? B? C? [5 marks] (c) A test with a sensitivity of 95% and a specificity of 90% has been developed for infection A. The test is only applied to people who report symptoms. (i) What is the probability that a person testing negative doesn't have infection A? [2.5 marks] (ii) What is the probability that there will be a screening error based on this test? [2.5 marks]