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
