Page 25 - Statistics and Data Analysis in Geology

Statistics and Data Analysis in Geology - Chapter 2

use a numerical scale, as for example a percentage scale. If we state that the chance
of rain tomorrow is 30%, then we imply that the chance of it not raining is 70%.
Scientists usually express probability as an arbitrary number ranging from 0 to
1, or an equivalent percentage ranging from 0 to 100%. If we say that the probability
of rain tomorrow is 0, we imply that we are absolutely certain that it will not rain.
If, on the other hand, we state that the probability of rain is 1, we are absolutely
certain that it will. Probability, expressed in this form, pertains to the likelihood
of an event. Absolute certainty is expressed at the ends of this scale, 0 and 1, with
different degrees of uncertainty in between. For example, if we rate the probability
of rain tomorrow as 1 /2 (and therefore of no rain as 1 /2), we express our view with
a maximum degree of uncertainty; the likelihood of rain is equal to that of no rain.
If we rate the probability of rain as 3/4 (1/4 probability of no rain), we express a
smaller degree of uncertainty, for we imply that it is three times as likely to rain as
it is not to rain.
Our estimates of the likelihood of rain may be based on many different factors,
including a subjective “feeling” about the matter. We may utilize the past behavior
of a phenomenon such as the weather to provide insight into its probable future be-
havior. This “relative frequency” approach to probability is intuitively appealing to
geologists, because the concept is closely akin to uniformitarianism. Other meth-
ods of defining and arriving at probabilities may be more appropriate in certain
circumstances. In carefully prescribed games of chance, the probabilities attached
to a specific outcome can be calculated exactly by combinatorial mathematics; we
will use this concept of probability in our initial discussions because of its relative
simplicity. An entire branch of statistics treats probabilities as subjective expres-
sions of the “degree of belief” that a particular outcome will occur. We must rely on
the subjective opinions of experts when considering such questions as the proba-
bility of failure of a new machine for which there is no past history of performance.
The subjective approach is widely used (although seldom admitted to) in the as-
sessment of the risks associated with petroleum and mineral exploration, where
relative-frequency based estimates of geologic conditions and events are difficult
to obtain (Harbaugh, Davis, and Wendebourg, 1995). The implications contained
in various concepts of probability are discussed in books by von Mises (1981) and
Fisher (1973). Fortunately, the mathematical manipulations of probabilities are
identical regardless of the source of the probabilities.
The chance of rain is a discrete probability; it either will or will not rain. A
classic example of discrete probability, used almost universally in statistics texts,
pertains to the outcome of the toss of an unbiased coin. A single toss has two
outcomes, heads or tails. Each is equally likely, so the probability of obtaining a
head is 1/2. This does not imply that every other toss will be a head, but rather
that, in the long run, heads will appear one-half of the time. Coin tossing is, then,
a clear-cut example of discrete probability. The event has two states and must
occupy one or the other; except for the vanishingly small possibility that the coin
will land precisely on edge, it must come up either heads or tails.
An interesting series of probabilities can be formed based on coin tossing. If
the probability of obtaining heads is 1/2, the probability of obtaining two heads in
a row is 1/2 . 1/2 = 1/4. Perhaps we are interested in knowing the probabilities of
obtaining three heads in a row; this will be 1/2 . 1/2 - 1/2 = 1/8. The logic behind
this progression is simple. On the first toss, our chances are 1/2 of obtaining a
head. If we do, our chances of obtaining a second head are again 1/2, because the

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