The Value of Information 545that 30 of the 100 sites tested good and proved to contain oil. The other entries
have similar interpretations. Loosely speaking, there is a correlation between
the test and the actual outcomes, demonstrated by the preponderance of cases
lying on the main diagonal of the table: Good tests (G) are likely to be associ-
ated with wet sites (W) and bad tests (B) with dry sites (D). However, there are
a significant number of false reports (G&D and B&W). The test results there-
fore are far from perfect.
Let’s use the historical frequencies in the table as an easy way to develop a
number of probabilities essential for evaluating the seismic test option. First,
note that the overall frequency of wet sites is 40 out of 100, or 40 percent. (See
the total at the bottom of the column labeled “Wet.”) Thus, this past record is
consistent with the initial probability assessment of the site under considera-
tion. Second, it is natural to inquire as to the chances of striking oil if the site
has tested good or, alternatively, if it has tested bad. Looking at the first row of
the table, we find that among 50 sites that tested good, 30 also turned out to
be wet. The notation Pr(WƒG) is used to denote the probability that the site is
wet given (or conditional on) a good test. From the table, we find that Pr(WƒG)
30/50 .6. Alternatively, if the test is bad, what are the chances of finding oil?
Of the 50 sites that tested bad, 10 were wet. Therefore, we have Pr(WƒB)
10/50 .2.
Let’s review what the table is telling us. Before taking the test, the best esti-
mate of the chance of striking oil is Pr(W) .4. This usually is termed the prior
probability(i.e., before new information is obtained). After taking the test, the
partners will revise their probability assessment based on the test outcome. One
of two conditional probabilitieswill be relevant. The initial assessment is revised
upward after a good test, Pr(WƒG) .6, and downward after a bad result,
Pr(WƒB) .2. Another important piece of data in the table is that 50 out of 100
sites tested good and 50 tested bad. That is, the probability that a site will test
good is .5.
One other point should be made. As presented, Table 13.1 lists the num-
ber of cases in each cell. By placing a decimal point before each entry, we
give the cells a slightly different interpretation. Now each is understood to be
a frequency or probability. For instance, the upper-left entry becomes .3; that
is, 30 percent of all sites tested good and proved to be wet. We use the nota-
tion Pr(W&G) .3 to denote the probability of this joint outcome. Similar
interpretations and notation hold for the other entries. This new interpreta-
tion has no effect on the conditional probabilities found earlier. For exam-
ple, the chance that the site is wet after a bad test is Pr(WƒB) .1/.5 .2,
exactly the same as the preceding result. Because of its flexibility and wide
application, we employ a probabilistic interpretation in the remainder of this
chapter.
It is important to see how the seismic information can improve the part-
ners’ decision. Figure 13.2 makes this point by depicting the new decisionc13TheValueofInformation.qxd 9/26/11 11:02 AM Page 545