You have one hour and twenty minutes to complete the exam Th

You have one hour and twenty minutes to complete the exam. This is a closed book exam. However, you are allowed to use a 8.5 Times 11 help sheet during the exam. Please show your all work! Let A and B be events with P(A) = 0.5, P(Bc) = 0.4, P(Ac Bc ) = 0.3. Calculate P(A B), P(A B), P(A B\\(Ac Bc)) and P(A\\Bc). Are A and B independent? Justify your answer.

Solution

P(A)=0.5, P(B^c) = 0.4, Therefore P(B) = 1-0.4 = 0.6. P(A^c B^c) = 0.3.

a) Therefore, P(A U B) = 1-0.3 = 0.7

P(A B) = P(A) + P(B) - P(A U B) = 0.5+0.6-0.7 = 0.4

P(A B | A^c U B^c) = P( (A B) ( A^c U B^c) / P( A^c U B^c)

P( A^c U B^c) = 1-P(A B) = 0.6

Therefore, P( (A B) ( A^c U B^c) = 0

Therefore, P(A B | A^c U B^c) = 0

P(A|B^c) = P(AB^c)/P(B^c) = (P(A) - P(A B))/P(B^c) = (0.5-0.4)/0.4 = 0.25

b) A and B will be independent if:

P(A B) = P(A)*P(B)

P(A B) = 0.4

P(A)*P(B) = 0.5*0.6 = 0.3

Therefore, A and B are not independent