Yesterday, Integrated Learning posted a 700 level GMAT question on our blog. Today, they have followed up with the answer:
This is a dependent probability problem. If you want to find the probability of choosing 2 black marbles, you will need to figure out the probability that the first marble will be black and that the second marble will be black. In this case, the question wants to know if that probability is larger than 1/3.
As you can see, when less than half the marbles are white, the probability of choosing 2 black marbles can be higher or lower than 1/3, depending on how many black marbles there are. This is not sufficient.
Statement 2 tells us that the probability of choosing one black marble and one white marble is 7/15. This is a trap. Since the probability given is exact, it may seem that only one scenario of black marbles and white marbles will work. If you work through all the scenarios, you will see that when there are 7 black marbles and 3 white marbles, the probability of choosing one of each is 7/15. However, it would also be true in reverse: If there were 7 white marbles and 3 black marbles, the probability would also be 7/15. Therefore, this is not enough information.
Combining them does give us enough information. From statement 2 we know that there must be 7 of one color and 3 of the other, and from statement 1 we know that there must be more black than white, so we know there must be 7 black marbles and 3 white marbles.
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