Chapter 18. Probability and Statistics
The need to make decisions on the basis of incomplete data is very widespread
in human life. We need to decide how warmly to dress and whether to carry an
umbrella when we leave home in the morning. We may have to decide whether to
risk a dangerous but potentially life-saving medical procedure. Such decisions rely
on statistical reasoning. Statistics is a science that is not exactly mathematics. It
uses mathematics, in the form of probability, but its procedures are the inverse ones
of fitting probability distributions to real-world data. Probability theory, on the
other hand, is a form of pure mathematics, with theorems that are just as certain
as those in algebra and analysis. We begin this chapter with the pure mathematics
and end with its application.
1. Probability
The word probability is related to the English words probe, probation, prove, and
approve. All of these words originally had a sense of testing or experimenting,^1
reflecting their descent from the Latin probo, which has these meanings. In other
languages the word used in this mathematical sense has a meaning more like plau-
sibility,^2 as in the German Wahrscheinlichkeit (literally, truth resemblance) or the
Russian veroyatnost' (literally, credibility, from the root ver-, meaning faith). The
concept is very difficult to define in declarative sentences, precisely because it refers
to phenomena that are normally described in the subjunctive mood. This mood
has nearly disappeared in modern English; it clings to a precarious existence in
the past tense, "If it were true that..." having replaced the older "If it be true
that...". The language of Aristotle and Plato, however, who were among the first
people to discuss chance philosophically, had two such moods, the subjunctive and
the optative, sometimes used interchangeably. As a result, they could express more
easily than we the intuitive concepts involved in discussing events that are imagined
rather than observed.
Intuitively, probability attempts to express the relative strength of the feeling
of confidence we have that an event will occur. How surprised would we be if
the event happened? How surprised would we be if it did not happen? Because
we do have different degrees of confidence in certain future events, quantitative
concepts become applicable to the study of probability. Generally speaking, if an
event occurs essentially all the time under specified conditions, such as an eclipse
(^1) The common phrase "the exception that proves the rule" is nowadays misunderstood and mis-
used because of this shift in the meaning of the word prove. Exceptions test rules, they do not
prove them in the current sense of that word. In fact, quite to the contrary, exceptions disprove
rules.
(^2) Here is another interesting word etymology. The root is plaudo, meaning strike, but specifically
meaning to clap one's hands together, to applaud. Once again, approval is involved in the notion
of probability.
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