Question

An oil company purchased an option on land in Alaska. Preliminary geologic studies assigned the following prior probabilities.

P(high-quality oil) = 0.50
P(medium-quality oil) = 0.20
P(no oil) = 0.30
If required, round your answers to two decimal places.

(a) What is the probability of finding oil?
(b) After 200 feet of drilling on the first well, a soil test is taken. The probabilities of finding the particular type of soil identified by the test are as follows.
P(soil | high-quality oil) = 0.20
P(soil | medium-quality oil) = 0.80
P(soil | no oil) = 0.20
How should the firm interpret the soil test?
The probability of finding oil is good. Given the probability of finding good soil, the oil company is more likely to find
- Select your answer -
oil.
What are the revised probabilities?
Let S = Soil test results
Events P(Ai) P(S | Ai) P(Ai ∩ S) P(Ai | S)
High Quality (A1)


Medium Quality (A2)


No Oil (A3)


P(S)=

What is the new probability of finding oil?

Answer 1

The probability of finding oil initially is 0.70. After a soil test, using Bayes' Theorem, the revised probabilities suggest an increased probability of finding oil, which is now 0.81.

Step-by-step explanation:

The student has been provided with prior probabilities of discovering oil in Alaska, and seeks assistance with calculating the probability of finding oil and revising probabilities after receiving soil test results while drilling.

Part A: Probability of Finding Oil

To calculate the probability of finding oil, we sum the probabilities of finding high-quality oil and medium-quality oil:

P(finding oil) = P(high-quality oil) + P(medium-quality oil)
= 0.50 + 0.20
= 0.70

Part B: Interpreting Soil Test Results and Revising Probabilities

Upon finding a certain type of soil, the revised probabilities need to be found using Bayes' Theorem. We calculate the joint probabilities and then the posterior probabilities:

P(A1 ∩ S) = P(soil | high-quality oil) × P(high-quality oil)

= 0.20 × 0.50

= 0.20 × 0.50 = 0.10

P(A2 ∩ S) = P(soil | medium-quality oil) × P(medium-quality oil)

= 0.80 × 0.20

= 0.16

P(A3 ∩ S) = P(soil | no oil) × P(no oil)

= 0.20 × 0.30

= 0.06

The marginal probability of the soil test is then:

P(S) = P(A1 ∩ S) + P(A2 ∩ S) + P(A3 ∩ S)
= 0.10 + 0.16 + 0.06
= 0.32

The revised posterior probabilities are:

P(A1 | S) = P(A1 ∩ S) / P(S)

= 0.10 / 0.32

≈ 0.31

P(A2 | S) = P(A2 ∩ S) / P(S)

= 0.16 / 0.32

≈ 0.50

The new probability of finding oil after the soil test is:

P(finding oil | S) = P(A1 | S) + P(A2 | S)

≈ 0.31 + 0.50
= 0.81