Page 36 - Statistics and Data Analysis in Geology

Elementary Statistics

the corresponding unconditional probabilities (the probability of finding a fossil,
given that the terrain is igneous, is much lower than the unconditional probability
of finding a fossil). Obviously, geologists exploit conditional probabilities in all
phases of their work, whether this is done consciously or not.
The relationship between conditional and unconditionai probabilities can be
expressed by Bayes’ theorem, named for Thomas Bayes, an eighteenth century En-
glish clergyman who investigated the manner in which probabilities change as more
information becomes available. Bayes’ basic equation is:

p(A,B) = p(BIA)p(A) (2.7)
which states that p(A, B), the joint probability that both events A and B occur, is
equal to the probability that B will occur given that A has already occurred, times
the probability that A will occur. p(BIA) is a conditional probability because it
expresses the probability that B will occur conditional upon the circumstance that
A has already occurred. If events A and B are related (or dependent), the fact that
A has already transpired tells us something about the likelihood that B will then
occur. Conversely, it is also true that

Therefore, the two can be equated, giving

which may be rewritten as

This is a most useful relationship, because sometimes we know one form of con-
ditional probability but are interested in the other. For example, we may deter-
mine that mining districts often are characterized by the presence of abnormal
geomagnetic fields. However, we are more interested in the converse, which is the
probability that an area will prove to be mineralized, conditional upon the pres-
ence of a magnetic anomaly. We can gather estimates of the conditional probabil-
ity p (anomaly I mineralization) and the unconditional probability p (mineralization)
from studies of known mining districts, but it may be more difficult to directly es-
timate p (mineralization I anomaly) because this would require the examination of
geomagnetic anomalies that may not yet have been prospected:
If there is an all-inclusive number of events Bi that are conditionally related to
event A, the probability that event A will occur is simply the sum of the conditional
probabilities p(AIBi) times the probabilities that the events Bi occur. That is,

If Equation (2.9) is substituted for p(A) in Bayes’ theorem, as given in Equation
(2.8), we have the more general equation

(2.10)

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