Question

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

Let $E_1,E_2,E_3$ be three events such that , $E_1=$ The location with earth composition was $C_1$

$E_2=$ The location with earth composition was $C_2$

$E_3=$ The location with earth composition was $C_3$

Also let $A$ be an event such that , $A=$ finding oil at the location

Then $P(E_1)=0.5,P(E_2)=0.3,P(E_3)=0.2$ and $P(A/E_1)=0.2,P(A/E_2)=0.4,P(A/E_3)=0.3$

Now we have to find $P(E_1/A).$

According to Bayes' Theorem we know that , $P(E_1/A)=\frac{P(E_1).P(A/E_1)}{P(E_1).P(A/E_1)+P(E_2).P(A/E_2)+P(E_3).P(A/E_3)}$

$\therefore$ The required probability is $P(E_1/A)=\frac{(0.5×0.2)}{(0.5×0.2)+(0.3×0.4)+(0.2×0.3)}=\frac{10}{28}=\frac{5}{14}$