A 30-question multiple choice test (say, a section from the College Boards) gives five choices for each answer. You lose 1/4 as much credit for each wrong answer as you gain for each right answer. (Problems left unanswered neither gain nor lose you credit.) If you guess at random on questions for which you don t know the answer, what is the probability that you will raise your total score?

Medical tests are not perfect. The standard tine test for tuberculosis attempts to identify carriers people who have been infected by the tuberculin bacteria. However, 8% of carriers test negative (a false negative ) and 4% of noncarriers test positive (a false positive ). Suppose you take the tine test and get a positive result. What is the probability that you are a carrier? (A carrier need not be sick. The bacteria may have been killed off naturally, or they may be in a latent state, capable of causing serious illness later. However, it is very difficult to distinguish between these two cases, so identified carriers are usually treated.)