Question

1.       In a basin area where oil is likely to be found underneath the surface, there are 3 locations with 3 different types of earth compositions, say C1, C2, & C3. The probabilities for these 3 compositions are 0.5, 0.3, & 0.2 respectively. Further, it has been found from the past experience that after drilling of well at these locations, the probabilities of finding oil is 0.2, 0.4, & 0.3 respectively. Suppose, a well is drilled at a location, & it yields oil, what is the probability that the earth composition was C1?

Answer 1

$P(C_1) = 0.5 \\ P(C_2) = 0.3 \\ P(C_3) = 0.2$

Let E denotes the event of finding the oil. Then

$P(E|C_1) = 0.2 \\ P(E|C_2) = 0.4 \\ P(E|C_3) = 0.3$

Let E^c denotes the event of not finding the oil.

Probability (oil is found):

$P(E) = P(E|C_1)P(C_1) + P(E|C_2)P(C_2)+P(E|C_3)P(C_3) \\ = (0.2 \times 0.5)+(0.4 \times 0.3)+(0.3 \times 0.2) \\ = 0.28$

The probability that the earth composition was C1:

$P(C_1|E) = \frac{P(C_1) \times P(E|C_1)}{P(E)} \\ = \frac{0.5 \times 0.2}{0.28} \\ = 0.357$

Answer: 0.357