Question $3(10$pts$)$ Suppose $0\mathrm{.}5\%$ of the population is infected with the COVID$-19$ virus, and an RT$-$PCR test is developed for COVID$-19.$ The RT$-$PCR test gives $3\%$ false positives $($giving a positive result for a person not infected with COVID$-19)$ and $2\%$ false negatives $($giving a negative result for a person infected with COVID$-19).$ What is the probability that Alice $($a random person$)$ tests positive? Alice just got the bad news that the test came back positive; what is the probability that Alice has the COVID$-19?$