Suppose that there is a new COVID - 19 test that produces a true positive result in an infected patient 95 % of the time and produces a false positive result in a healthy patient 4 % of the time . Suppose also that 1 % of the population is currently infected with COVID - 19 .

( a ) Suppose you have taken the test once and received a positive result . Use Bayes ' Rule to find the conditional probability that you are infected , given that you received a positive result . Round your answer to the nearest ten - thousandths place .

( b ) Now suppose that you were to take the same test TWICE ( you can assume that the results of both tests are independent to each other ) . What is the conditional probability that BOTHI tests produce true positive results given that you are infected ? What is the conditional probability that BOTH tests produce false positive results given that you are healthy ?

( c ) Using the probabilities you found in ( b ) , use Bayes ' Rule again to find the conditional probability that you actually are infected , given that you tested positive in BOTH tests . Round your answer to the nearest ten thousandths place .

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