9781118041581

Once we have the table of joint probabilities, it is a simple matter to compute

the probabilities needed for the decision tree. The probability of a given test

result—say, Pr(G)—is found by adding across the appropriate row. In algebraic

terms,

[13.2]

so that Pr(G)  .3  .2  .5. Note that a good test can occur when the site is

really wet andwhen the site is really dry.

Next, we calculate revised probabilities. The chance that the site is wet

given a good seismic test is computed as

[13.3]

so that Pr(WƒG)  .3/.5  .6. Similarly, we have Pr(WƒB) Pr(W&B)/Pr(B) 

.1/.5  .2. Of course, these are precisely the answers we found earlier from the

joint probability table. But in this case, the partners did not begin with the table

in front of them; rather, they started with a prior probability, Pr(W), and with

information on the accuracy of the test, Pr(GƒW) and Pr(BƒD). From these facts,

they were able to calculate the necessary probabilities: Pr(G), Pr(WƒG), and

Pr(WƒB).

Pr(WƒG)

Pr(W&G)

Pr(G)

,

Pr(G)Pr(W&G)Pr(D&G),

548 Chapter 13 The Value of Information

CHECK

STATION 1

Suppose the partners face the same seismic test just discussed but are less optimistic

about the site; the prior probability now is Pr(W) .28. Construct the joint probability

table, and compute Pr(WƒƒG) and Pr(WƒƒB).

Bayes’ Theorem

With a little practice, the step-by-step mechanics of calculating revised proba-

bilities become routine. In fact, the sequence of steps can be condensed. For

example, if we replace Pr(W&G) in Equation 13.3 with the right-hand side of

Equation 13.1, we obtain

[13.4]

This equation is the most common form of Bayes’ theorem(named after

Reverend Thomas Bayes, who wrote an essay on the subject in 1763). Bayes’ the-

Pr(WƒG)c

Pr(GƒW)

Pr(G)

d[Pr(W)]

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