Suppose a college student decides to take a series of n tests at once, to diagnose whetherthey have a certain disease (any individual test is not perfectly reliable, so they hope toreduce their uncertainty by taking multiple tests. At least they’re free!). Let D be the eventthat they have the disease, p = P (D) be the prior probability that they have the disease,and q = 1 − p. Let Tj be the event that they test positive on the jth test.(a) Assume for this part that the test results are conditionally independent giventhe student’s disease status. Let a = P (Tj |D) and b = P (Tj |Dc), where a and b don’tdepend on j. Find the conditional probability that they have the disease, given thatthey test positive on all n of the n tests.Note: Your answer should be an expression involving a, b, p, and/or q.(b) Suppose that they test positive on all n tests. However, some people have acertain gene that makes them always test positive. Let G be the event that they havethe gene. Assume that P (G) = 1/2 and that D and G are independent. If they donot have the gene, then the test results are conditionally independent given his diseasestatus. Let a0 = P (Tj |D ∩ Gc) and b0 = P (Tj |Dc ∩ Gc), where a0 and b0 don’t dependon j. Find the conditional probability that they have the disease, given that they testedpositive on all n of the tests.Note: Your answer should be an expression involving a, b, a0, b0, p, and/or q
Suppose a college student decides to take a series of n tests at once, to diagnose whether
they have a certain disease (any individual test is not perfectly reliable, so they hope to
reduce their uncertainty by taking multiple tests. At least they’re free!). Let D be the event
that they have the disease, p = P (D) be the prior
and q = 1 − p. Let Tj be the event that they test positive on the jth test.
(a) Assume for this part that the test results are conditionally independent given
the student’s disease status. Let a = P (Tj |D) and b = P (Tj |Dc), where a and b don’t
depend on j. Find the conditional probability that they have the disease, given that
they test positive on all n of the n tests.
Note: Your answer should be an expression involving a, b, p, and/or q.
(b) Suppose that they test positive on all n tests. However, some people have a
certain gene that makes them always test positive. Let G be the event that they have
the gene. Assume that P (G) = 1/2 and that D and G are independent. If they do
not have the gene, then the test results are conditionally independent given his disease
status. Let a0 = P (Tj |D ∩ Gc) and b0 = P (Tj |Dc ∩ Gc), where a0 and b0 don’t depend
on j. Find the conditional probability that they have the disease, given that they tested
positive on all n of the tests.
Note: Your answer should be an expression involving a, b, a0, b0, p, and/or q
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