Data Sufficiency

Suppose A and B are two events, not independent. Is the probability P(A and B) > 1/3?

A. P(A)=0.8 and P(B)=0.7
B. P(A or B) = 0.9

If A and B are independent, then P(A and B) = P(A) * P(B)

If A and B are NOT independent, then P(A and B) = P(A) * P(B|A). (The second term is "the probability of B given that A happens", and can be written as P(B) * P(A|B) / P(A).)

S1:

We know that the probability of both A and B is LESS THAN OR EQUAL TO the probability of only the less likely one happening. (Getting A and B is less likely than getting B.) So

P(A) > P(B) ≥ P(A and B)

We also know that P(A and B) is MORE LIKELY than P(A) + P(B) - 1. This is because we know that

1 ≥ P(A or B) = P(A) + P(B) - P(A and B)

which we can rearrange by writing

1 ≥ P(A) + P(B) - P(A and B)

and adding P(A and B) to both sides, and subtracting 1 from both sides:

P(A and B) ≥ P(A) + P(B) - 1

So we have the inequality:

.7 ≥ P(A and B) ≥ (.8 + .7 - 1)

.7 ≥ P(A and B) ≥ .5

So P(A and B) ≥ .5, which is > (1/3): SUFFICIENT!

S2:

We don't know the relationship between A and B here. For instance, suppose A and B are not independent in one sense: if A happens, then B only has a 10% chance of happening, and if A doesn't happen, then B has a 0% chance of happening. In this case, P(A and B) < (1/3). But if A and B are "almost independent" (that isn't logically right, but you get the idea), then it's easy to generate numbers in which P(A or B) = 0.9 and P(A and B) > (1/3). So we don't know.

Algebraically, we don't have great limits here. We know from the previous explanation that

P(A) + P(B) - P(A and B) = P(A or B)

so

P(A) + P(B) - P(A and B) = 0.9

but this doesn't allow us to isolate any further.

Just as an addendum, most of the probability properties used in my solution seem well beyond the known scope of the GMAT, so I doubt a question like this is going to appear on your exam. (The test has tested dependent probability in the abstract before, but AFAIK only in the sense that P(A) * P(B|A) > P(A) * P(B).)

Dependent and conditional probability in the abstract are quite difficult theoretically and practically, and enough students struggle with probability already that even basic, independent probability questions seem to be rated quite difficult by the testmakers.