EXAMPLE 1 For this question, the most common response by far is librarian,
followed by farmer and airline pilot. Apparently the psychological sketch fits
the commonly perceived stereotype of a librarian. Overlooked in this answer
is one crucial fact: The individual has been picked at random from the labor
force. This being the case, one’s prior probability (before reading the sketch)
should be heavily weighted toward salesperson. Salespeople comprise roughly
15 percent of the labor force; farmers are next, at under 3 percent; and librar-
ians comprise only a fraction of 1 percent. How much should the sketch alter
these prior probabilities? Surely very little, since we have but two sentences
about Steve, and they are not very informative or discriminating. Perhaps half
of all persons might be described as orderly and passionate about detail. Up to
a quarter of the population might regard themselves as “shy.” Moreover, not all
librarians are shy, nor are all salespersons gregarious. In short, the observation
that the worker has been picked at random is the overriding determinant of his
likely occupation. Nonetheless, most people overlook this fact and invest too
much confidence in the relatively uninformative sketch.EXAMPLE 2 The nearly unanimous answer to this question is 50 percent. One
reasons that the draw rules out the gold-gold box, leaving either the silver-
silver or gold-silver boxes as equally likely. Despite its overwhelming intuitive
appeal, this answer is wrong. The chances are two in three that the other coin
will be silver. An easy way to see this is to note that there are a total of three sil-
ver coins in the boxes, and the coin you see is equally likely to be any of the
three. But two of these coins reside in the all-silver box, meaning its neighbor
is silver. Only one of the silver coins has a gold neighbor. Thus, upon seeing a
silver coin, the odds are two to one against the other coin being gold. Bayes’
theorem provides a neat confirmation of this correct answer:On the right-hand side, the first term is 1.0 (a silver coin is a certainty from the
SS box), the second term is .5 (the overall chance of picking a silver coin is
1/2), and the last term (the prior chance of picking the SS box at random) is
1/3. Thus, we find Pr(SS boxƒS) 2/3.EXAMPLE 3 On the basis of the near-perfect test, most respondents see can-
cer as very likely, in the range of 50 to 95 percent. However, the correct chance
is only about 2 percent. This surprising answer can be confirmed by using
Bayes’ theorem or applying the following simple reasoning. Suppose 1,000
59-year-old men were to be tested. According to the prior probability, one man
actually would have cancer; with near certainty, he would test positive. Of the
remaining 999 healthy men, 95 percent would test negative. But 5 percent, or
50 men, would record false positives. In all, one would expect 51 positives, 1Pr(SS boxƒS)cPr(SƒSS box)
Pr(S)dPr(SS box).556 Chapter 13 The Value of InformationAnswersc13TheValueofInformation.qxd 9/26/11 11:02 AM Page 556