Bunuel wrote:
Suppose A and B are two events that are not independent. Is the probability P(A and B) > 1/3?
(1) P(A) = 0.8 and P(B) = 0.7
(2) P(A or B) = 0.9
Kudos for a correct solution.
MAGOOSH OFFICIAL SOLUTION:Statement #1: this is very tricky. There is no cut-and-dry probability rule for this. we have to think about overlap. The total probability space, which encompasses anything that possibly could happen, has a size of 1, and P(A) and P(B) have to fit in this space. These two have a size of 0.8 and 0.7 respectively, so they are going to overlap. Think about it visually —
Attachment:
gdspqop_img4.png [ 5.18 KiB | Viewed 7589 times ]
Push the P(A) = 0.8 all the way to the left (whatever that means!), leaving the 0.2 outside of A on the right. Now, push P(B) = 0.7 all the way to the right, leaving the 0.3 outside of B on the left. Suppose the 0.2 outside of A is inside B, and the 0.03 outside of B is inside A. This would be the minimum possible overlap, and even then the overlap, P(A and B), equals 0.5. Thus, P(A and B) ≥ 0.5, so it must be greater than 1/3. this statement allows us to give a definitive answer to the prompt question. This statement, alone and by itself, is sufficient.
Statement #2: forget everything we learned in the analysis of statement (1). Now, all we know is P(A or B) = 0.9, and we know absolutely nothing about P(A) or P(B). We can calculate nothing else. This statement, alone and by itself, is insufficient.
First sufficient, second not sufficient.
Answer = A.
_________________