In an article that appears on the website 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 given— say 99 in 100— are 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 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
i. P(TD| D)
iii. P(C)
ii. P(TD| C)
iv. P(D)
b. Use the probabilities from Part (a) to construct a “hypothetical 1000” table.
c. What is the value of P(TD)?
d. Use the table to compute the probability that a person is clean given that the test result is dirty, P(C| TD). Is this value consistent with the argument given in the quote? Explain.