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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) ¼ .4 and P(B) ¼ .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 tw

Respuesta :

Answer:

a) 0.75

b) 0.4375

c) 0.5714

Step-by-step explanation:

Solution:-

- The events are defined as follows:

Event A : The Asian project is successful

Event B : The European project is successful

- The two given events are independent. Their respective probabilities are:

 P ( A ) = 0.25

 P ( B ) = 0.25

- The conditional probability for European project to fail given that asian project also failed.

- The probability can be expressed as:

            P ( B ' / A ' ) = P ( A' & B' ) / P ( A' )

- According to the property of independent events we have:

            P ( A ' & B ' ) = ( 1 - P ( A ) )* ( 1 - P ( B ) )

Therefore,

            P ( B ' / A ' ) = [ ( 1 - P ( A ) )* ( 1 - P ( B ) ) ] /  ( 1 - P ( A ) )

            P ( B ' / A ' ) = 1 - P ( B )

Answer: The probability is simply the failure of event (B) : The european project fails = 0.75.

b) The probability that at-least one of the two projects will successful consists of (either A or B is successful) or  ( Both are successful). We can mathematically express it as:

          P ( At-least 1 project is success ) = P ( A U B ) + P ( A & B )

          P ( At-least 1 project is success ) = P ( A )*(1 - P ( B )) + P ( B )*(1 - P ( A )]+ P ( A ) * P ( B )

                                                                = 2*0.25*0.75 + 0.25^2

                                                                = 0.4375

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

- We can mathematically express the required conditional probability as follows with help of  part b):

           P ( A / At-least 1 project is success ) = P ( A & any at-least 1 is success) / P ( At-least 1 project is success )

- The probability of P ( A & any at-least 1 is success), consists of event A success and event B fails or both are a success:

          P ( A & any at-least 1 is success) = P ( A )*( 1 - P ( B ) ) + P ( A )*P ( B )

                                                                 = 0.25*0.75 + 0.25^2

                                                                 = 0.25

- The conditional probability can now be evaluated:

          = P ( A & any at-least 1 is success) / P ( At-least 1 project is success )

          = 0.25 / 0.4375

          = 0.5714

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