Consider a test which detects if a person has a disease.

R: outcome of the test on a person

D: whether the person actually has the disease

1: be the probability that the test is positive if the person has it

2: be the probability that the test is negative if the person doesn't have the disease.

The probability that it reports that someone has the disease (R = 1) when they actually do (D = 1), is 1, and the probability that it reports that someone doesn't have the disease when they don't is 0.

Formally: p(R = 1|D = 1) = 1 p(R = 0|D = 0) = 0

Prior probability of a person having this disease is p(D) = .

a) A patient has the test performed. Derive the posterior probability that the person actually has the disease in terms of 1, 0 and given that

1) the test comes back positive

2) the test comes back negative.

b) After the results of the first test come back positive, the doctor runs it a second time. This time it comes back negative. Derive the posterior probability that the person actually has the disease after this second round of testing in terms of 1, 0 and . If 1 = 0 what did we learn about whether the person has the disease?

c) Suppose 1 = 0 = .99 and = 0.001. Suppose 1000 patients get tested, and they're all negative. On average, how many of these patients actually have the disease? I.E., what is the expected number of false negatives?

d) The standard test for COVID-19 is the PCR test. It is estimated that it can have a false negative rate of around 15%, meaning that when a person who has COVID-19 takes the test, 15% of the time the test will indicate that they don't.

The test almost never indicates the presence of COVID-19 when a person doesn't have the disease; assume that this happens only 1% of the time. In early September it was estimated that among a random sample of 100,000 people in Canada approximately 50 have COVID-10 at any given time. If organizers of an event will require everyone to show a negative test before attending, how many people can there be at the event and still keep the probability of there being at least one infected person below 1%? What if you wanted to keep probability of at least one person being infected below .1%?