HW - THREE BAYES RULE PROBLEMS

1. EYEWITNESS TESTIMONY. A man was struck by a speeding taxi as he crossed the street. Aneyewitness testified the taxi (which did not stop) was blue. The man sued the Blue Cabcompany for his medical expenses. The city has only two cab companies, Blue Cab and GreenCab. Green Cab has 85% of all taxis in the city.At the trial, the man's lawyer showed that the eyewitness is 80% reliable in identifying thecolor of taxis. That is, the eyewitness was able to identify the color of a taxi correctly 80% ofthe time, under conditions similar to those the night of the accident. The lawyer concludesthat it is extremely likely that the man was hit by a Blue Cab taxi.Given the eyewitness's testimony, what is the probability the man was hit by a Blue Cab taxi?

2.MEDICAL TESTS. Medical tests are used to indicate whether a person has a particulardisease.The sensitivity of a test is the probability that a person having the disease will test positive(indicating the person has the disease, i.e., it's the probability the test will pick up the factthat a person has the disease).The specificity of a test is the probability that a person not having the disease will testnegative.A test for a certain disease has been in use for many years. Based on the "track record" ofthis test, its sensitivity is 0.934 and its specificity is 0.968. Finally, in the population, 1 in 500people has the disease.a. What is the probability that a person who tests positive actually has the disease?b. What value would the specificity of the test have to have in order for the probability toexceed 0.5 that a person who tests positive actually has the disease?