Probability and statistics

Question

A company is trialling a new COVID test. They find that the probability that the test is positive given that the patient does have COVID is 0.94, whereas the probability that the test is positive given that they don’t have COVID is 0.03.  In each case give answers to 4 decimal places.

a) Suppose the test is used in a population where it is estimated

that 4% of people tested have COVID. What is the probability

of a test being negative?

(b) In the same population as above, what is the probability that

someone does not have COVID if the test is negative?

(c) Now the test is used in a population with an unknown infection

rate. If 10% of this population tests positive, what percent of the

population is likely to actually have COVID?

Answer 1

This is the Bayesian statistics

a) combine true positive and false positive, subtract from 1

1 - (0.94 * 0.04 + 0.03 * (1 - 0.04)) = 0.934

b) negative test is in two cases: true negative and false negative

true negative: ans from (a) - false negative:

0.93 - 0.04 * (1 - 0.94) = 0.928

conditional true negative if tested negative = 0.928 / 0.934 = 0.994

c) x is is the infections rate, combine true positive and false positive

0.94 * x + 0.03 * (1 - x) = 0.1

0.91 * x = 0.07

x = 0.077