Geoscience Reference
In-Depth Information
2.2 Probability Calculus and Discrete
Frequency Distributions
Kolmogorov (
1931
) generally is credited with establishing the axioms of
mathematical statistics. His formal definitions that involve Borel sets are beyond
the scope of this topic. However, the rules presented in this section are in agreement
with Kolmogorov's original axioms.
Geoscientists should have a basic understanding of probabilities and how
to calculate them. The following example provides an illustration of rules of
multiplication and addition of probabilities. Suppose that wildcats in a sedimentary
basin have a success ratio of
p
0.2. One then can answer questions like: What is
the probability that 0, 1 or 2 wildcats will strike oil if two new wells are drilled?
Writing these probabilities as
p
(0),
p
(1) and
p
(2),
¼
the answers are
p
(0)
¼
p
)
2
p
2
(1
0.04, respectively. The
sum of these three probabilities is 1. The probability that one or two wildcats will
strike oil is
p
(1) +
p
(2)
¼
0.64,
p
(1)
¼
2
p
(1
p
)
¼
0.32 and
p
(2)
¼
¼
¼
0.36.
2.2.1 Conditional Probability and Bayes' Theorem
A slightly more difficult problem and its solution are as follows: Suppose that
p
(
D
|
B
) represents the conditional probability that event
D
occurs given event
B
(
e.g.
, a mineral deposit
D
occurs in a small unit cell underlain by rock type
B
on a geological map). This conditional probability obeys three basic rules
(
cf
. Lindley
1987
, p. 18):
1. Convexity: 0
p
(
D
|
B
)
1;
D
occurs with certainty if
B
logically implies
D
;
1, and
p
(
D
c
|
B
)
0 where
D
c
represents the complement of
D
;
then,
p
(
D
|
B
)
¼
¼
2. Addition:
p
(
B
[
C
|
D
)
¼
p
(
B
|
D
)+
p
(
C
|
D
)
p
(
B
\
C
|
D
); and
3. Multiplication:
p
(
B
\
C
|
D
)
¼
p
(
B
|
D
)·
p
(
C
|
B
\
D
).
These three basic rules lead to many other rules. For example, replacement of
B
by
B
\
D
in the multiplication rule gives:
p
(
B
\
C
|
D
)
¼
p
(
B
|
D
)·
p
(
C
|
B
\
D
). Like-
wise, it is readily derived that:
p
(
B
\
C
\
D
)
¼
p
(
B
|
D
)·
p
(
C
|
B
\
D
)·
p
(
D
). This leads
to Bayes' theorem in odds form:
pD
B
pB
C
pD
C
pD
c
C
\
C
\
D
pD
c
B
pB
C
¼
D
c
\
C
\
or
OD
B
OD
C
¼
\
C
exp
W
B\C
ð
Þ
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