A certain virus infects one in every 300 people. The test has a Sensitivity of 8

Question

A certain virus infects one in every 300 people. The test has a Sensitivity of 80%. This is the true positive rate, and implies a false negative rate of 20%. The test has a Specificity of 90%. This is the true negative rate, and implies a false positive rate of 10%. Let A be the event "the person is infected" and B be the event "the person tests positive". Hint: Use the "fake population approach" - Imagine a large city with, say 12,000,000 people. Figure out how many have the disease, how many don't, and for each, how many test positive and how many test negative. Then use the "restricted sample space" approach. a) Find the probability that a person has the virus given that they have tested positive, i.e.

A) Find P(A|B). Round your answer to the nearest tenth of a percent and do not include a percent sign.

B) Find the probability that a person does not have the virus given that they test negative, i.e. find P(A'|B'). Round your answer to the nearest tenth of a percent and do not include a percent sign.

Explanation / Answer

a) here P(B) =P(A)*P(B|A)+P(Ac)*P(B|Ac) =(1/300)*0.8+(299/300)*0.1 =0.1023

hence  probability that a person has the virus given that they have tested positive =P(A|B)=P(A)*P(B|A)/P(B)

=(1/300)*0.8/0.1023 =0.0261

b)

here P(B') =1-P(B) =1-0.0261 =0.8977

hence P(A'|B') =P(A')*P(B'|A')/P(B') =(299/300)*0.9/0.8977 =0.9993

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