# Bayes' Theorem Application

Ontario Public Health conducted a study on their test results for detecting SARS-CoV-2 (the virus that causes COVID-19) from Jan-April 20201. These are all patients who had symptoms and went to get tested. They had a set of 569 patients test positive on multiple tests, which we assume means they had the virus. Now, 85 tested negative on their first test but subsequently tested positive multiple times later. To measure the probability of someone having the disease in general, epidemiologists use the percentage positive (also called \positive test rate" or "prevalence") number which is the number of tests that come back positive, in general, in a region. For Ontario, the prevalence right now is 7%. As of now the prevalence for Florida is 18.8% (hint: use this as your probability of being infected in each region as if it is from a random sample of the population as a whole). Scientists have also found that about the false positive rate for the test is around 1%, that is, 1% of the time the test comes back true for healthy people. Answer the following questions about this data, be sure to first define random variables:

1. Given this data, and a uniform probability distribution over the set of patients, what is the probability of a false negative, that is of the event V , that the patient is actually infected with the virus, yet they show a negative test result on their first test?

Here's my solution:

A is the event that a patient has the virus B is the event that a patient tests negative on their first test

$$P(A|B) = P(B|A)P(A)/P(B)$$

where $$P(B) = P(B|A)P(A) + P(B|A')P(A')$$

and $$P(A) = 0.07$$, $$P(B|A) = 85/569$$, $$P(B|A') = 0.99$$, $$P(A') = 0.93$$

How do I incorporate the uniform random variable in the solution? Is this solution correct for Ontario?