Page 30 - Statistics and Data Analysis in Geology

Elementary Statistics

Number of discoveries
Figure 2-3. Discrete distribution giving the probability of making n discoveries in a five-hole
drilling program when the success ratio (probability of a discovery) is 10%.

exploration company is determined to discover two new fields in a virgin basin it
is prospecting, and will drill as many holes as required to achieve its goal. We can
investigate the probability that it will require 2,3,4,. . ., up to n exploratory holes
before two discoveries are made. The same conditions that govern the binomial
distribution may be assumed, except that the number of “trials” is not fixed.
The probability distribution that governs such an experiment is called the neg-
ative binomial, and its development is very similar to that of the binomial distribu-
tion. As in that example, p is the probability of a discovery and Y is the number of
“successes” or discovery wells. However, n, the number of trials, is not specified.
Instead, we wish to find the probability that x dry holes will be drilled before Y
discoveries are made. The negative binomial has the form

Note the similarity between this equation and Equation (2.3); the term r + x - 1 ap-
pears because the last hole drilled in a sequence must be the rth success. Expanding
Equation (2.4) gives
(Y fX - l)!
P= (1 - pIXpY
(Y - l)!x!

If the regional success ratio is assumed to be lo%, the probability that a two-
hole exploration program will meet the company’s goal of two discoveries can be
calculated: (2 + 0 - l)!
P= * (1 - 0.1O)O . o.102
(2 - l)!O!
--. o.90° o.102
l!
-
1!0!
= 1 ’ 1 * 0.01 = 0.01
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