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# GMAT Question of the Day - Data Sufficiency - Probability

Roy has a bowl with 12 red and j blue marbles. If Roy picks two random marbles out of the bowl it is more likely that he will pick one of each color than only two blue marbles. Are there an equal number of red and blue marbles in the bowl?

(1) j > 11

(2) The chance of picking two blue marbles is greater than 5/23

## GMAT Question of the Day Solution

On GMAT Data Sufficiency questions take your time to understand the given information. Avoid rushing into the statements. In this case you need to unravel this statement: If Roy picks two random marbles out of the bowl it is more likely that he will pick one of each color than only two blue marbles. This is critical. Now you might be thinking that this question is very open ended because you don't know how many blue marbles you have but the given information severely limits the options. The number of blue marbles must be less than or equal to 25. Here's the math to calculate that.

This is based on this statement: If Roy picks two random marbles out of the bowl it is more likely that he will pick one of each color than only two blue marbles.

j/(12+j) * (j-1)/(12+j-1) *(2!/2!)  < j/(12+j)*12/(12+j-1)*(2!/(1!*1!))

j<25

Statement (1) Could be a Yes, 12, or a no with anything greater than 12. Statement (2) give you the same information but in a slightly different way. We need to know whether we have 12 blue marbles so just try plugging in 11 marbles or 12 marbles see which one (or maybe both) agrees with the information in the statement. We can see that when picking 11 you get exactly 5/23. So there must be more than 11 blue marbles. That is the same info as statement 1. You could have the same number of marbles 12 and 12 or you could have more blue marbles, up to 24.

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