5.3 Exercise 15

In a region in the Gulf Coast area, oil deposits are highly likely to be associated with underground salt domes. If H is the event that an oil deposit is present in an area, and S is the event of a salt dome in the area, experience indicates $P \left(\right. S \left|\right. H \left.\right) = 0 . 9$ and $P \left(\right. S \left|\right. H^{c} \left.\right) = 0 . 1$. Company executives believe the odds favoring oil in the area is at least 1 in 10. It decides to conduct two independent geophysical surveys for the presence of a salt dome. Let $E_{1} , E_{2}$ be the events the surveys indicate a salt dome. Because the surveys are tests for the geological structure, not the presence of oil, and the tests are carried out in an operationally independent manner, it seems reasonable to assume $\left\{H , E_{1} , E_{2}\right\} ci \left|\right. S$ and $ci \left|\right. S^{c}$. Data on the reliability of the surveys yield the following probabilities

Determine the posterior odds $\frac{P \left(\right. H \left|\right. E_{1} E_{2} \left.\right)}{P \left(\right. H^{c} \left|\right. E_{1} E_{2} \left.\right)}$. Should the well be drilled?