Question

suppose that the probability we can find the virus in a single test is no better than flipping a coin, which is 50%. Suppose we conduct a series of independent test using the same patient's sample, and make the decision that the patient is contracted with the virus. If any one of these test is positive. How many independent tests do we need to carry out such that such a scheme give us a false positive rate (probability that a healthy patient is diagnosed a sick) below 0.1%?

Answer 1

Let the number of tests be "N". Now if anyone of these tests is positive then we say that the patient is contracted with the virus i.e. if we observe a positive test on or before the Nth test then we say that the patient is contracted with the virus. Thus the false positive rate is given by:

i.e.N follows a geometric distribution with parameter 0.5. Now, We want this to be less than 0.1. Thus we'll try this for different N's. For N=1 F=0.5.

For N=2, F=0.25

For N=3, F=0.125

For N=4, F=0.0625

Thus N=4 is the answer.