Bayes rule:Pr(A|X)=[pr(X|A)*pr(A)]/[pr(X|A)*pr(A)+pr(X|B)*pr(B)]

Instructions:

Remember that pr(A) and pr(B) are prior probabilities of A and B.In our example, we can treat A as actually having had Covid and B as actually not having had Covid. We base these numbers on the population prevalence of Covid. They need to add up to 1, as you have either had or not had it. There is no other option.

Also, pr(X|A) and pr(X|B) are conditional probabilities, where X is the likelihood of a positive test result for Covid antibodies. There are concerns noted in the media that the level of false positives of these antibodies tests are too high. This tutorial will show you why this might be a problem. Pr(X|A) is the likelihood that the test will say you've had Covid(X) when you've actually had Covid(A)(we'll assume this is 90%). Pr(X|B) is the false positive rate: the likelihood that the test will say you've had Covid(X) when you haven't actually had Covid(i.e. when B is true).

QUESTION 1

Based on the instructions, calculate the posterior probability i.e. the likelihood that you actually had Covid, given that you received a positive Covid antibody test result, assuming the following:

True positive likelihood = 90% (this is a guess!)

False positive likelihood = 15% (there are some reports suggesting that this could be as high as this!)

Population incidence of Covid = 20% (this is a made-up number)

1. 0.6
2. 0.7
3. 0.8
4. 0.5

QUESTION 2

Now, they say that a good antibody test should have a far lower false positive rate, like 5% or even 2%. Let's see what difference this would make to the test's ability to predict: based on the instructions, calculate the posterior probability i.e. the likelihood that you actually had Covid, given that you received a positive Covid antibody test result, assuming the following:

True positive likelihood = 90% (this is a guess!)

False positive likelihood = 5% (this is the conservative estimate of how the test should be)

Population incidence of Covid = 20% (this is a made up number)

1. 0.61
2. 0.72
3. 0.82
4. 0.5

QUESTION 3

Now, they say that a good antibody test should have a far lower false positive rate, like 5% or even 2%. Let's see what difference this would make to the test's ability to predict: based on the instructions, calculate the posterior probability i.e. the likelihood that you actually had Covid, given that you received a positive Covid antibody test result, assuming the following:

True positive likelihood = 90% (this is a guess!)

False positive likelihood = 2% (this is the more optimistic estimate of how the test should be)

Population incidence of Covid = 20% (this is a made up number)

1. 0.92
2. 0.82
3. 0.72
4. 0.62