Consider two medical tests, A and B for a virus. Test A is 90% effective at recognizing the virus when it is present, but has a 5% false positive rate. Test B is 80% effective at recognizing the virus when it is present, but only has a 1% false positive rate. The two tests are independent (i. e., they use different means for identifying the virus). The virus is carried by 2% of all people. If you could use only one of the two tests to identify the virus, which would you choose? Justify your answer mathematically. How much more certain can you be if you can use both tests?

Question

Mathematics

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1 April, 03:29

Consider two medical tests, A and B for a virus. Test A is 90% effective at recognizing the virus when it is present, but has a 5% false positive rate. Test B is 80% effective at recognizing the virus when it is present, but only has a 1% false positive rate. The two tests are independent (i. e., they use different means for identifying the virus). The virus is carried by 2% of all people. If you could use only one of the two tests to identify the virus, which would you choose? Justify your answer mathematically. How much more certain can you be if you can use both tests?

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Ana Tanner · Answer 1

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P (V|A) is not 0.95. It is opposite:

P (A|V) = 0.95

From the text we can also conclude, that

P (A|∼V) = 0.1

P (B|V) = 0.9

P (B|∼V) = 0.05

P (V) = 0.01

P (∼V) = 0.99

What you need to calculate and compare is P (V|A) and P (V|B)

P (V∩A) = P (A) ⋅P (V|A) ⇒P (V|A) = P (V∩A) P (A)

P (V∩A) means, that Joe has a virus and it is detected, so

P (V∩A) = P (V) ⋅P (A|V) = 0.01⋅0.95=0.0095

P (A) is sum of two options: "Joe has virus and it is detected" and "Joe has no virus, but it was mistakenly detected", therefore:

P (A) = P (V) ⋅P (A|V) + P (∼V) ⋅P (A|∼V) = 0.01⋅0.95+0.99⋅0.1=0.1085