Exercise $3.5$ P$.$Consider two medical tests, A and B$,$ for a virus. Test A is $95\%$ effective at recognizing the virus

when it is present, but has a $10\%$ false positive rate $($indicating that the virus is present, when it is not$).$ Test

B is $90\%$ effective at recognizing the virus, but has a $5\%$ false positive rate. The two tests use independent

methods of identifying the virus. The virus is carried by $1\%$ of all people. Say that a person is testsed for

the virus using only one of the tests, and that the test comes back positive for carrying the virus. Which test

returning positive is more indicative of someone really carrying the virus? Justify your answer mathematically.

Exercise $4.$ Suppose you are a witness to a nighttime hit$-$and$-$run accident involving a taxi in Athens. All

taxis in Athens are blue or green. You swear, under oath, that the taxi was blue. Extensive testing shows that,

under dim lighting conditions, discrimination between blue and green is $75\%$ reliable.

a$)$ Is it possible to calculate the most likely color for the taxi? $($Hint: distinguish carefully between the $5$ P$.$

proposition that the taxi is blue and the proposition that it appears blue.$)$

b$)$ What if you know that $9$ out of $10$ Athenian taxis are green?