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Patients in two demographics were tested for a disease. The probability that a patient has the disease if the test is positive is $0.95$ in both demographics. The probability that a patient has the disease and the test is negative is $.05$ in both demographics. The probability of a positive test in demographic 1 is $.08$ and the probability of a positive test in demographic 2 is $.05$. What is the probability of the disease in demographic 1?

I'm not sure where to start, and I'm ending up with 3 variables. Is this the correct direction? When I have three variables, how do I know where to put the bars and commas? i.e. $P(A,B|C)$ vs $P(A|B,C)$ vs $P(A,B,C)$ Also, any hints as to how to proceed?

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  • $\begingroup$ The statement "The probability of a patient that has a disease and a positive test for the disease is $0.95$ in both demographics," is unclear about what it is supposed to be measuring. $\endgroup$ – Graham Kemp Dec 3 '17 at 7:26
  • $\begingroup$ rephrased to "The probability that a patient has the disease if the test is positive is $0.95$" $\endgroup$ – NikNik Dec 3 '17 at 13:24
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Using Sum Rule:

$P(D|Dem1) = P(D|T,Dem1)P(T|Dem1) + P(D|T',Dem1)P(T'|Dem1)$

= $(0.95)(.08) + (0.05)(1-.08) = 0.122$

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