A medical test classifies subject as being either positive (infected with virus)

Question

A medical test classifies subject as being either positive (infected with virus) or negative (not infected). The test is believed to be 70% accurate. In other words, if an infected person is tested, he or she has 70% probability of the test being positive; while if a healthy (non-infected) person is tested, there is 70% probability of the test being negative. Suppose that 10% of all people are infected and 90% are healthy. If a person tests positive, what is the probability that person is actually infected? (Hint: Let T = {a randomly selected person tests positive} and I = {a randomly selected person is infected with the virus}. Note that P(T|I) = P(not T|not I) = 0.70. Also, P(I) = 0.10.)

Explanation / Answer

the problem can be solved using Bayes theoreem

A be the evnt that a person is infected P(A)=.10

B be the evnt that a person is not infected P(B) = .90

C be the evnt that a person tests positive

P(C/A)=.7 and P(C/B)=.3

we have to find P(A/C) = (P(A)*P(C/A))/ (P(A)*P(C/A)+P(B)*P(C/B)) = (.1*.7)/((.1*.7)+(.9*.3))=0.20588

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