Independent event does not seem to follow rule

[thisischris]

Hello. I'm trying to understand why P(A|B) ≠ (PA) given that the events are indepent is this case. I am not sure if my working out is correct but the answer is 1/8 in the answer section.

Events A and B are defined in the sample space S. The events A and B are independent.
P(A) = 0.3, P(B) = 0.4 and P(A U B) = 0.65.

Find P(A|B).


3. The Attempt at a Solution :
P(A n B) = P(A) + P(B) - P(A U B)
P(A n B) = 0.3 + 0.4 - 0.65 = 0.05

P(A|B) = P(A n B) / P(B)
P(A|B) = 0.05 / 0.4
P(A|B) = 1 / 8

 

[Ray Vickson]

Hello. I'm trying to understand why P(A|B) ≠ (PA) given that the events are indepent is this case. I am not sure if my working out is correct but the answer is 1/8 in the answer section.

Events A and B are defined in the sample space S. The events A and B are independent.
P(A) = 0.3, P(B) = 0.4 and P(A U B) = 0.65.

Find P(A|B).


3. The Attempt at a Solution :
P(A n B) = P(A) + P(B) - P(A U B)
P(A n B) = 0.3 + 0.4 - 0.65 = 0.05

P(A|B) = P(A n B) / P(B)
P(A|B) = 0.05 / 0.4
P(A|B) = 1 / 8

There is something wrong with the problem statement. From P(A)=0.3, P(B)=0.4 and P(A or B) =0.65, it follows that P(A & B) = 0.05, just as you have said. However, P(A)*P(B) = 0.12, which is ≠ P(A & B), so A and B cannot be independent.

RGV

 

[thisischris]

There is something wrong with the problem statement. From P(A)=0.3, P(B)=0.4 and P(A or B) =0.65, it follows that P(A & B) = 0.15, just as you have said. However, P(A)*P(B) = 0.12, which is ≠ P(A & B), so A and B cannot be independent.

RGV

Thank you. Must be an error I'm guessing.