The Problem:
For each of the following cases, indicate what, if anything, is wrong with the protagonist's reasoning. If there is a problem (and there may not be), suggest how to carry out a more careful analysis. Your answers must be expressed in terms of the principles of probabilistic reasoning. Assume that all statements of fact, no matter where they come from, are true. If there is a problem, it lies in the statement in bold face.
Case 1: Psychiatric Diagnosis
Dr. Wrawsharck, a clinical psychologist, tested a person who had been referred for diagnosis. The person had been arrested on a charge of assault and battery, had pleaded an insanity defense, and was referred by the courts prior to the trial. As part of the assessment Wrawsharck gave the client a projective ink blot test. For many of the ink blots, the client claimed to see mutilated bodies and assorted body parts. "This kind of response", thought Dr. Wrawsharck, "Is found in over 90% of people who suffer from paranoid schizophrenia". In his report to the court he wrote "There is a high probability that the defendant is a paranoid schizophrenic" .
Case 2: The Lie Detector
Sam Shade ran a detective agency and lie detection consultancy. He was asked to investigate a suspected case of corporate espionage. Sally Solomon was suspected of passing privileged information to her company's rivals. Sam set up a lie detector test for Sally, in which she was asked about her contacts with the other company. Sally failed the lie detector test. "I have done careful research into the validity of my lie detection procedures", said Sam. "I find that 85% of all people who are lying fail my lie detector test. Therefore, Sally was probably lying".
Case 3: Wine Tasting
Adrienne claimed to be an expert on wines. To test her claim, Francois challenged her to identify a certain Chateau Lafitte in a blind tasting. While Adrienne was blind-folded, he set up glasses of four red wines, including the Chateau Lafitte and three foils. Adrienne took a sip of each, and pointed to the Lafitte. "Lucky guess!", said Francois, "Do it again". Five more times (six times in all) he set up a blind tasting, with different foils and a different target wine each time (a Saint-Emilion, a Sauternes, and so on). Each time Adrienne picked out the target wine. "If I were guessing, the chances of my getting all six right would be less than one in 4,000", said Adrienne.
Case 4: Amazing Lottery Success
A few years ago a New Jersey woman won two lotteries within four months. Newspapers widely reported it to be an extraordinary coincidence, a one-in-seventeen-trillion (17,000,000,000,000) chance, based on the assumption that there were over 4 million (4,000,000) tickets sold for each lottery. [Hint: Imagine that the woman in problem was Freda Smith of 273 Dock St,. Hoboken NJ, who won lotteries decided on January 20, 1992 and May 12, 1992]
Case 5: Fighting Terror
A man who planned to travel to a dangerous location was concerned with the possibility of there being of a bomb on his plane. He determined the probability to be low (roughly one in a thousand), but not low enough to make him comfortable. So he decided to travel with a bomb in his suitcase. He reasoned that the probability of two bombs being on board is infinitesimal (one in a thousand thousand). [Assume that the man is not a terrorist, does not know any terrorists, and the TSA staff is unable to detect the bomb]
General Hint: All probabilities can be expressed as the probability of an uncertain event, conditional on some known or assumed piece of information. In each problem, one or more probabilities are given or implied. What is the event that each of these probabilities refers to? The statement in bold face is a probability statement. Is this probability used in an appropriate way?