Page 29 - Statistics and Data Analysis in Geology

Statistics and Data Analysis in Geology - Chapter 2

7. However, the (n - Y) dry holes and the Y discoveries may be arranged in
(:) combinations or, equivalently, in n!/(n - Y)!Y! different ways. So, the

probability that Y discoveries will be made in a drilling program of n wildcats
is n!
P= (1 - p)n-rpr
(n - Y)! Y!
This is an expression of the binomial distribution, and gives the probability
that Y successes will occur in n trials, when the probability of success in a
single trial is p.
The binomial equation can be solved to determine the probability of occurrence
of any particular combination of successes and failures, for any desired number of
trials and any specified probability. These probabilities have already been com-
puted and tabulated for many combinations of n, Y, and p. Using either the equa-
tion or published tables such as those in Hald (1952), many interesting questions
can be investigated. For example, suppose we wish to develop the probabilities
associated with a five-hole exploration program in a virgin basin where the suc-
cess ratio is anticipated to be about 10%. What is the probability that the entire
exploration program will be a total failure, with no discoveries? Such an outcome
is called “gambler’s ruin” for obvious reasons, and the binomial expression has the
terms
n=5
Y=O
p = 0.10
p = (0 .o.ioo . (1 - 0.10)’

5!
--. 1 * 0.90’
-
5!0!
= 1 1 . 0.59 = 0.59
0
The probability that no discoveries will result from the exploratory effort is almost
60%.
If only one hole is a discovery, it may pay off the costs of the entire explo-
ration effort. What is the probability that one well will come in during the five-hole
exploration campaign?

p = (3) .o.io1. (1 - 0.10)4

’!
=-. 0.10. 0.904
4!1!
= 5 . 0.10 * 0.656 = 0.328

Using either the binomial equation or a table of the binomial distribution, the prob-
abilities associated with all possible outcomes of the five-hole drilling program can
be found. These are shown in Figure 2-3.
Other discrete probability distributions can be developed for those experimen-
tal situations where the basic assumptions are different. Suppose, for example, an

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