knight247 wrote:Suppose A and B are two events, 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

OA is A

My query is just this. Since the stem explicitly states that the two events are not independent, are we to assume that the two events ARE dependent?

If that's the case, shouldn't P(A)*P(B)= ZERO? I mean, the probability of getting heads and tails on a the same coin toss is ZERO, so how can these two non-independent events have a positive probability?

Similarly, with statement (2). Considering both events are not independent, can't we do a trial-and-error on the values of P(A) and P(B) and figure out whether or not P(A)*P(B)<0.333?

If we take the two values as 0.45 and 0.45 OR 0.3 and 0.6 or any other values, we always get values that are less than 0.333, so I'm wondering what is incorrect with that method?

Detailed explanations would be appreciated. Thank you.

Dear knight247,
I'm happy to respond. I wrote this problem: it's question #7 on this blog:
https://magoosh.com/gmat/2013/gmat-data- ... obability/

My friend, you are profoundly confusing the independent/not-independent distinction with the disjoint/non-disjoint distinction. Those are two completely different distinctions in probability, and "not independent" certainly does not mean "disjoint," but you are treating it this way.

You will see a discussion here.
https://magoosh.com/gmat/2012/gmat-math- ... ity-rules/

Here's a brief explanation. Disjoint means mutually exclusive. Events A & B are disjoint if one happening prevents the other from happening. On a single coin toss, head & tails are disjoint. If A is "the NY Mets win the World Series in 2015" and B is "the LA Dodgers win the World Series in 2015," then those two events are disjoint, because they can't both happen. If Event A is "I am in the USA" and Event B is "I am in Kazakhstan," then the events are disjoint, because they can't both be true simultaneously: if one is true, it makes the other false. If A & B are disjoint, then P(A and B) = 0, and P(A or B) = P(A) + P(B). If they are not disjoint, then P(A or B) = P(A) + P(B) - P(A and B).

By contrast, independent means when one happens, it has no influence on whether the other happens. When two events are not independent, there may be a causal relationship, or there many be some weak but discernible channel of influence. Two coin flips are truly independent: one has zero influence on the other. If I draw a card from a shuffled deck, look at it, replace it, re-shuffle, and draw again, the two draws are independent. If I draw once from a shuffled deck, and without replacement, draw a second card, that's dependent, because if I drew, say, the five of hearts on my first draw, that makes drawing another heart or drawing another 5 less likely on the second draw. Being a corporate CEO and being female are not independent: in a world of perfect gender equality, they would be independent, but in our current world, yes, there are some female CEOs, but they are in the minority, and the vast majority are males. If the event "being a CEO" and the event "being a female" were truly independent, then when we picked a random CEO, the probability of picking a female CEO would be 50%, the same percent as picking a female in the general population. If A and B are independent, then P(A and B) = P(A)*P(B).

You see, "being female" and "being a CEO" are not independent, because there are definitely many fewer female CEOs then there would be in a truly just world, but they certainly are not disjoint, because there definitely are some female CEOs out there. Many real-world categories involving human beings are simultaneously "not independent" and "not disjoint."

It's very important to understand these distinctions.

Does all this make sense?
Mike