Page 33 - Statistics and Data Analysis in Geology

Statistics and Data Analysis in Geology - Chapter 2

for objects located randomly in space. The geometric distribution is a special case
of the negative binomial, appropriate when interest is focused on the number of
trials prior to the initial success. The multinomial distribution is an extension of
the binomial where more than two mutually exclusive outcomes are possible. These
topics are extensively developed in most books on probability theory, such as those
by Parzen (1960) or Ash (1970).
An important characteristic of all of the discrete probability distributions just
discussed is that the probability of success remains constant from trial to trial.
Statisticians discuss simple experiments called sampling with replacement in
which this assumption holds strictly true. A typical experiment would involve an
urn filled with red and white balls; if a ball is selected at random, the probability it
will be red is equal to the proportion of red balls originally in the urn. If the ball is
then returned to the urn, the proportions of the two colors remain unchanged, and
the probability of drawing a red ball on a second trial remains unchanged as well.
The probability also will remain approximately constant if there are a very large
number of balls in the urn, even if those selected are not returned, because their
removal causes an infinitesimal change in the proportions among those remaining.
This latter condition usually is assumed to prevail in many geological situations
where discrete probability distributions are applied. In our binomial probability
example, the “urn” consists of the geologic basin where exploration is occurring,
and the red and white balls correspond to undiscovered reservoirs and barren areas.
As long as the number of undrilled locations is large, and the number of prospects
that have been drilled (and hence “removed from the urn”) is small, the assump-
tion of constant probability of discovery seems reasonable. However, if a sampling
experiment is performed with a small number of colored balls initially in the urn
and those taken from the urn are not returned, the probabilities obviously change
with each draw. Such an experiment is called sampling without replacement, and
is governed by the discrete hypergeometric distribution. Geologic problems where
its use is appropriate are not common, but McCray (1975) presents an example
from geophysical exploration for petroleum.
In some circumstances it is possible to know the size of the population within
which discoveries will be made. Suppose an offshore concession contains ten well-
defined seismic features that seem to represent structures caused by movement of
salt at depth. From experience in nearby offshore tracts, it is believed that about
40% of such seismic features will prove to be productive structures. Because of
budgetary limitations, it is not possible to drill all of the features in the current
exploration program. The hypergeometric distribution can be used to estimate the
probabilities that specified numbers of discoveries will be made if only some of the
identified prospects are drilled.
The binomial distribution is not appropriate for this problem because the prob-
ability of a discovery changes with each exploratory hole. If there are four reser-
voirs distributed among the ten seismic features, the discovery of one reservoir
increases the odds against finding another because there are fewer remaining to
be discovered. Conversely, drilling a dry hole on a seismic feature increases the
probability that the remaining untested features will prove productive, because
one nonproductive feature has been eliminated from the population.
Calculating the hypergeometric probability consists simply of finding all of the
possible combinations of producing and dry features within the population, and
then enumerating those combinations that yield the desired number of discoveries.

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