9781118041581

Other Applications 555

must be able to explain the reasons for their forecasts to others. Again, formal

analysis is essential. To take an extreme case, how confident would you be in a

forecaster, no matter how accurate the track record, if he or she confessed to

using astrological tables or a Ouija board?

The second difficulty is that forecasts based informally on intuition, judg-

ment, and experience frequently are inaccurate or biased. For instance, a com-

mon layperson’s belief is that a large head, forehead, or brain is a sign of

intelligence. But scientific evidence shows this hypothesis to be false. Perhaps

the best way to understand the difficulties in making probabilistic predictions

is to test yourself on some short (but subtle) examples.

For each of the following examples, use your informal judgment to come

up with your own best probability estimate of the event in question. After

recording your intuitive responses, you may wish to use a formal method (a

joint probability table or Bayes’ theorem) to find solutions. Keep yourself hon-

est by writing down your responses before turning to the answers that follow.

EXAMPLE 1 An individual, picked at random from the U.S. labor force, is

described in the following short psychological sketch:

Steve is shy and withdrawn, with little interest in people or the world of

reality. He has a need for order and structure and a passion for detail.

Which of the following is Steve’s most likely occupation: (1) farmer, (2) sales-

person, (3) librarian, (4) airline pilot, or (5) doctor?

EXAMPLE 2 You are presented with three boxes. Each box has two com-

partments. In one box there is a gold coin in each compartment. In the sec-

ond, there is a silver coin in each compartment. In the third, there is a gold

coin in one compartment and a silver coin in the other. The compartments

are closed, and the boxes (identical from the outside) are randomly mixed.

You choose one box and are allowed to open one compartment. Suppose you

see a silver coin. What are the chances that the coin in the other compart-

ment is silver?

EXAMPLE 3 During his annual medical examination, a 59-year-old man had

a chemical test on a sample of his stool. Blood in the stool is a possible indi-

cation of cancer of the bowel. This cancer is relatively rare; for a man this

age, the incidence of bowel cancer is about 1 in 1,000. It is also quite curable

provided it is identified early, while still small. In answer to his questions, the

man is told that a test is available and is 95 percent accurate; that is, if can-

cer is present, the test will be positive with 95 percent probability. Likewise,

if there is no cancer, the test will be negative 95 percent of the time. Suppose

the test result is positive. What is the chance that the man has cancer of the

bowel?

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