Page 37 - Statistics and Data Analysis in Geology

Statistics and Data Analysis in Geology - Chapter 2

A simple example involving two possible prior events, B1 and B2, will illustrate
the use of Bayes’ theorem. A fragment of a hitherto unknown species of mosasaur
has been found in a stream bed in western Kansas, and a vertebrate paleontologist
would like to send a student field party out to search for more complete remains.
Unfortunately, the source of the fragment cannot be identified with certainty be-
cause the fossil was found below the junction of two dry stream tributaries. The
drainage basin of the larger stream contains about 18 mi2, while the basin drained
by the smaller stream includes only about 10 mi2. On the basis of just this infor-
mation alone, we might postulate that the probability that the fragment came from
one of the drainage basins is proportional to the area of the basin, or

10
p(B2) = - = 0.36
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However, an examination of a geologic report and map of the region discloses the
additional information that about 3 5% of the outcropping Cretaceous rocks in the
larger basin are marine, while almost 80% of the outcropping Cretaceous rocks in
the smaller basin are marine. We may therefore postulate the conditional prob-
ability that, given a fossil is derived from basin Bi, it will be a marine fossil, as
proportional to the percentage of the Cretaceous outcrop area in the basin that is
marine, or for basin B1
p(AIB1) = 0.35
and for basin BZ
p(AIB2) = 0.80

Using these probabilities and Bayes’ theorem, we can assess the conditional
probability that the fossil fragment came from basin B1, given that the fossil is
marine.

-
- (0.35) (0.64)
(0.35) (0.64) + (0.80) (0.36)
= 0.44
Similarly, the probability that the fossil came from the smaller basin is

= 0.56
Fortunately for the students who must search the area, it seems somewhat
more likely that the fragment of marine fossil mosasaur came from the smaller
basin than from the larger. However, the differences in probability are very small
and, of course, depend upon the reasonableness of the assumptions used to esti-
mate the probabilities.

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