An oil exploration company currently has two active projects, one in Asia and th

Question

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) = 0.6 and P(B) =0s. a) If the Asian project is not successful, what is the probability that the European project is also not successful? Explain yourasoning. O since the events are independent, then A' and B' are independent. O since the events are independent, then A' and B' are mutually exclusive. O Since the events are not independent, then A' and B' are mutually exclusive. O Since the events are independent, then A' and B' are not independent. (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? (Round your answer to three decimal places.)

Explanation / Answer

(a)

This is equivalent to saying that both the projects are unsuccessful.

This is denoted by the event P(A' and B')

Since A and B are independent, so

P(A' and B') = P(A')*P(B') = (1-P(A))*(1-P(B)) = (1-0.6)*(1-0.5) = 0.2

Since the events are independent, then A' and B' are independent.

(b)

Using complement rule:

P(at least one project is successful) = 1 - P(both projects are unsuccessful)

As calculated above, P(both projects are unsuccessful) = 0.2

So,

P(at least one project is successful) = 1-0.2 = 0.8

(c)

Let C denote the event that atleast one of the projects is unsuccessful.

We have:

P(C) = 0.8

Here we are asked to calculate the probability P(only Asian project is successful | C)

Using conditional probability formula:

P(only Asian project is successful | C) = P(only Asian project is successful and C)/P(C)

Also by intuitive reasoning you can see that:

P(only Asian project is successful and C) = P(only Asian project is successful)

Now,

P(only Asian project is successful) = P(Asian project is successful and European is unsuccessful) = P(A and B') = P(A)*P(B') = 0.6*(1-0.5) = 0.3

So,

P(only Asian project is successful given that atleast one project is successful) = P(only Asian project is successful)/P(C) = 0.3/0.8 = 0.375

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