Question

1. An oil exploration company currently has two active projects, one in Asia and the other in Europe. Let A be the event that the Asian project is successful and B be the event that the European project is successful. Suppose that A and B are independent events with P(A)=0.4 and P(B)=0.7.

a. If the Asian project is not successful, what is the probability that the European project is also not successful? Explain your reasoning.

b. What is the probability that at least one of the two projects will be successful?  

c. Given that at least one of the two projects is successful, what is the probability that only the Asian project is successful?

Answer 1

Now here A and B are independent events.

Probability that Asian Project is successful is P(A) = 0.4

this implies probability that Asian Project is Unsuccessful is P(A') = 1 - P(A) = 1 - 0.4 = 0.6

Similarly, Probability that European project is successful is P(B) = 0.7

so, probability that European project is Unsuccessful is P(B') = 1 - 0.7 = 0.3

a) here we have to find probability that both the project are not successful.

i.e P( A' B' ) = P(A') X P(B')

As A and B are independent, A' and B' are also independent.

thus P( A' B' ) = 0.6 X 0.3

         P( A' B' ) = 0.18

Therefore, probability the both the project are not successful is 0.18.

b) to find the probability that at least one of the two projects will be successful

here there are 3 cases

1st: - A is successful but B is not successful.

2nd: - A is not successful but B is successful.

3rd: - Both A and B are successful.

so 1st Case,

P( A B' ) = P(A) X P(B')

P( A B' ) = 0.4 X 0.3

P( A B' ) = 0.12

Similarly,

2nd Case P( A' B ) = P(A') X P(B)

                  P( A' B ) = 0.6 X 0.7

                  P( A' B ) = 0.42

3rd Case, P( A B) = P(A) X P(B)

                P( A B) = 0.4 X 0.7

                P( A B) = 0.28

Now adding probabilities of all the cases, we get

probability that atleast one of the project will be successful is 0.12 + 0.42 + 0.28 = 0.82.

c) given that atleast one project is successful, what is the probability that only Asian project will be successful.

now here the probablitiy that only asian project will be successful is the 1st case in b) part.

i.e

P( A B' ) = P(A) X P(B')

P( A B' ) = 0.4 X 0.3

P( A B' ) = 0.12

now probability that atleast one project is successful is 0.82 (from part b).

thus, the probability that only Asian project will be successful, given that atleast one project is successful is

Hence,

given that atleast one project is successful, the probability that only Asian project will be successful is 0.15.