In an article that appears on the web site of the American Statistical Association (www. amstat. org), Carlton Gunn, a public defender in Seattle, Washington, wrote about how he uses statistics in his work as an attorney. He states: I personally have used statistics in trying to challenge the reliability of drug testing results. Suppose the chance of a mistake in the taking and processing of a urine sample for a drug test is just 1 in 100. And your client has a "dirty" (i. e., positive) test result. Only a 1 in 100 chance that it could be wrong? Not necessarily. If the vast majority of all tests givenâsay 99 in 100âare truly clean, then you get one false dirty and one true dirty in every 100 tests, so that half of the dirty tests are false.

Define the following events as given below:
TD = event that the test result is dirty
TC = event that the test result is clean
D = event that the person tested is actually dirty
C = event that the person tested is actually clean

a. Using the information in the quote, what are the values of:

1. P(TD|D)
2. P(C)
3. P(TD|C)
4. P(D)

b. Use the law of total probability to find P(TD) .
c. Use Bayesâ rule to evaluate P(C \ TD) . Is this value consistent with the argument given in the quote? Explain.