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 Post subject: GMAT Probability (Data Sufficiency)Posted: Wed Apr 17, 2013 2:54 am

Joined: Sun May 30, 2010 3:15 am
Posts: 424
A bag contains black and white marbles. How many white marbles are there in the bag?
(1) The probability of picking a white marble at random is 1/2.
(2) There are 10 black marbles in the bag.

A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
D. Either statement BY ITSELF is sufficient to answer the question.
E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(C) The probability equals to the number of favourable outcomes divided by the number of all possible outcomes. Letâ€™s denote the number of white marbles by w and the number of black marbles by b.

Statement (1) means w/(w + b) = 1/2. It yields w = b. So we can tell that white marbles make exactly half of all marbles in the bag. But we do NOT know how many black marbles or marbles in total there are. Therefore statement (1) by itself is NOT sufficient to answer the question.

Statement (2) means b = 10. But we do NOT know anything about white marbles. Statement (2) by itself is NOT sufficient.

If we use the both statements together, statement (1) implies w = b, while statement (2) is b = 10. Therefore w = 10. So statements (1) and (2) taken together are sufficient to answer the question, even though neither statement by itself is sufficient. The correct answer is C.
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Since marbles are identical. Probability of drawing a white marble will always be 1/2 in any draw.
In any draw of one marble it will be either a black marble or a white marble.

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 Post subject: Re: GMAT Probability (Data Sufficiency)Posted: Wed Apr 17, 2013 2:55 am

Joined: Sun May 30, 2010 2:23 am
Posts: 498
Quote:
Since marbles are identical. Probability of drawing a white marble will always be 1/2 in any draw.
In any draw of one marble it will be either a black marble or a white marble.
I would like you to pay attention to the word "equiprobable".

Suppose we have two marbles in a bag: one black and one white. We pick a marble at random. What outcomes do we have?
Outcome 1: we picked the white marble.
Outcome 2: we picked the black marble.
"at random" means that picking each marble is equiprobable. So P1 = P2, where P1 and P2 are the probabilities of the outcomes.
We also know that at least one of the marbles is chosen, so the probabilities add up to 1.
P1 + P2 = 1
From these two formulas we deduce that the probability of picking the white marble is 1/2.

Suppose we have three marbles in a bag: one white marble and two black marbles. We pick a marble at random. What outcomes do we have?
Let's number the marbles in the way that #1 is the white marble, and #2, #3 are the black marbles. The outcomes we have:
Outcome 1: we picked the marble #1 (white).
Outcome 2: we picked the marble #2 (black).
Outcome 3: we picked the marble #2 (black).
These outcomes are equiporbable and one of them will definitely happen. So we have the same formulas P1 = P2 = P3 and P1 + P2 + P3 = 1.
From the formulas we deduce that P1 = 1/3 (the probability of picking the white marble is 1/3).

Could we consider outcomes as
Outcome 1: we picked the white marble.
Outcome 2: we picked a black marble.
?

Yes, we could. But these outcomes will NOT be equiprobable. So you can not say that the probability of each outcome is 1/2. In fact, the outcome "we picked a black marble" consists of "we picked the black marble #2" and "we picked the black marble #3". Its probability is 1/3 + 1/3 = 2/3 .

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