Page 35 - Statistics and Data Analysis in Geology

Statistics and Data Analysis in Geology - Chapter 2

Number of successes
Figure 2-7. Discrete distribution for the probability of n discoveries in drilling four out of
ten prospects when four prospects contain oil.

two outcomes, hit porous sandstone and dril2 into an anticline, are not mutually ex-
clusive as we hope that both can occur simultaneously. Since the presence of a
sandstone is governed by factors that operated at the time of deposition, and since
the occurrence of an anticlinal fold is presumed to be related to tectonic conditions
at a later time, the two outcomes are unrelated, or independent. If two events are
not mutually exclusive but are independent, the joint probability that they will
occur simultaneously is the product of their separate probabilities of occurrence.
That is, p (hit sandstone and drill anticline) = p (hit sandstone) x p (drill anticline).
This is the muZtipZicative rule of probability.
Two events may be related in some way, so that the outcome of one is depen-
dent in part on the outcome of the other. The joint probability of such events is
said to be conditional. Such events are extremely important in geology, because
we may be able to observe one event directly, but the other event is hidden. If
the two are conditional, the occurrence of the observable event tells us something
about the likely state of the hidden event. For example, the upward movement
of magma in chambers beneath a volcano such as Mt. St. Helens in Washington is
believed to cause a harmonic tremor, a particular type of earthquake. We cannot
directly observe an active magma chamber, but we can observe and record the seis-
mic activity associated with a volcano. If a conditional relationship exists between
these two events, the occurrence of harmonic tremors may help predict eruptions.
If p(tremor) is the probability that a harmonic tremor occurs and p(eruption) is
the probability of a subsequent volcanic eruption, then p (tremor and eruption) #
p (tremor) x p (eruption) if the two events have a conditional relationship.
The conditional probability that an eruption will occur, given that harmonic
tremors have been recorded, is denoted p (eruption 1 tremor). In this instance the
conditional probability of an eruption is greater than the unconditional probability,
or p (eruption), which is simply the probability that an eruption will occur without
any knowledge of other events. Other conditional probabilities may be lower than

22