D: Omer has the disease being tested for. ⚫ Tj: Omer tests positive on the jth test for j = 1, . Let p = P(D) be the prior probability that he has the d 1. Assume for this part the test results are conditiona status. Let a = P(T; | D) and bo = P(T; | Dc] Find the posterior probability that Omer has the di all n of the n tests. Hint: Since ao does not depend on j (the index of lot since P(T; | D) P(T; | D) = a² for any test i 2. Suppose some people have a gene that makes ther medical test. Let G be the event that Omer has the that D and G are independent - that is, the gene susceptible to the disease.
D: Omer has the disease being tested for. ⚫ Tj: Omer tests positive on the jth test for j = 1, . Let p = P(D) be the prior probability that he has the d 1. Assume for this part the test results are conditiona status. Let a = P(T; | D) and bo = P(T; | Dc] Find the posterior probability that Omer has the di all n of the n tests. Hint: Since ao does not depend on j (the index of lot since P(T; | D) P(T; | D) = a² for any test i 2. Suppose some people have a gene that makes ther medical test. Let G be the event that Omer has the that D and G are independent - that is, the gene susceptible to the disease.
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By hand solve and focus on parts 2 and 3. Show all steps of the equations.
Transcribed Image Text:• D: Omer has the disease being tested for.
⚫ Tj: Omer tests positive on the jth test for j = 1, 2, ..., n.
Let p = P(D) be the prior probability that he has the disease.
1. Assume for this part the test results are conditionally independent given Omer's disease
status. Let a = P(T; | D) and b₁ = P(T; | Dc), where ao and bo don't depend on j.
Find the posterior probability that Omer has the disease, given that he tests positive on
all n of the n tests.
Hint: Since ao does not depend on j (the index of a test result), the algebra simplifies a
lot since P(T; | D) P(T; | D) = a² for any test indices i and j. The same holds for bo.
2. Suppose some people have a gene that makes them always test positive on this type of
medical test. Let G be the event that Omer has the gene. Assume that P(G) = ½ and
that D and G are independent — that is, the gene does not make you more or less
susceptible to the disease.
If Omer has the gene, he'll test positive on all n tests.
If Omer does not have the gene, then the test results are conditionally independent
given his disease status. Let a₁ = P(T; | D, GC) and b₁ = P(T; | Dc, GC), where a1
and bi don't depend on j.
Now, suppose that Omer tests positive on all n tests and find the posterior probability
that Omer has the disease.
3. Using the same setup as in part (2.), find the posterior probability that Omer has the
gene given that he tests positive on all n of the tests.
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