Sec. 1.3. MISUSES, MISCALCULATIONS, AND PARADOXES IN PROBABILITY 7

1.3 MISUSES, MISCALCULATIONS, AND PARADOXES IN PROBABILITY

The misuse of probability and statistics in everyday life is quite common. Many of the

misuses are illustrated by the following examples. Consider a defendant in a murder trial

who pleads not guilty to murdering his wife. The defendant has on numerous occasions

beaten his wife. His lawyer argues that, yes, the defendant has beaten his wife but that

among men who do so, the probability that one of them will actually murder his wife is

only 0.001, that is, only one in a thousand. Let us assume that this statement is true. It

is meant to sway the jury by implying that the fact of beating one’s wife is no indicator

of murdering one’s wife. Unfortunately, unless the members of the jury have taken a good

course in probability, they might not be aware that a far more signiﬁcant question is the

following: Given that a battered wife is murdered, what is the probability that the husband is

the murderer ? Statistics show that this probability is, in fact, greater than one-half.

In the 1996 presidential race, Senator Bob Dole’s age became an issue. His opponents

claimed that a 72-year-old white male has a 27 percent risk of dying in the next ﬁve years.

Thus it was argued, were Bob Dole elected, the probability that he would fail to survive his

term was greater than one-in-four. The trouble with this argument is that the probability

of survival, as computed, was not conditioned on additional pertinent facts. As it happens,

if a 72-year-old male is still in the workforce and, additionally, happens to be rich, then

taking these additional facts into consideration, the average 73-year-old (the age at which

Dole would have assumed the presidency) has only a one-in-eight chance of dying in the

next four years [1-3].

Misuse of probability appears frequently in predicting life elsewhere in the universe.

In his book Probability 1 (Harcourt Brace & Company, 1998), Amir Aczel assures us

that we can be certain that alien life forms are out there just waiting to be discovered.

However, in a cogent review of Aczel’s book, John Durant of London’s Imperial College

writes,

Statistics are extremely powerful and important, and Aczel is a very clear and capable

exponent of them. But statistics cannot substitute for empirical knowledge about the

way the universe behaves. We now have no plausible way of arriving at robust estimates

about the way the universe behaves. We now have no plausible way of arriving at

robust estimates for the probability of life arriving spontaneously when the conditions

are right. So, until we either discover extraterrestrial life or understand far more about

how at least one form of life—terrestrial life—ﬁrst appeared, we can do little more

than guess at the likelihood that life exists elsewhere in the universe. And as long as

we’re guessing, we should not dress up our interesting speculations as mathematical

certainties.

The computation of probabilities based on relative frequency can lead to paradoxes. An

excellent example is found in [1-3]. We repeat the example here:

In a sample of American women between the ages of 35 and 50, 4 out of 100 develop

breast cancer within a year. Does Mrs. Smith, a 49-year-old American woman, there-

fore have a 4% chance of getting breast cancer in the next year? There is no answer.

Suppose that in a sample of women between the ages of 45 and 90—a class to which