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## Official Guide Explanation:Data Sufficiency #122

Background

This is just one of hundreds of free explanations I've created to the quantitative questions in The Official Guide for GMAT Quantitative Review (2nd ed.). Click the links on the question number, difficulty level, and categories to find explanations for other problems.

These are the same explanations that are featured in my "Guides to the Official Guide" PDF booklets. However, because of the limitations of HTML and cross-browser compatibility, some mathematical concepts, such as fractions and roots, do not display as clearly online.

Question: 122
Page: 162
Difficulty: 6 (Moderately Difficult)
Category 1: Word Problems > Other >
Category 2: Arithmetic > Discrete Probability >

Explanation: Statement (1) is insufficient. If more than half of the employees are women, at least 6 of the employees must be women. If exactly 6 of the employees are women, then the probability that the first employee chosen is female is (6/10). After one woman has been chosen, the probability that the second employee chosen is female is (5/9). The probability that both are female, then, is the product: (6/10)((5/9)) = (3/9) = (1/3). That's less than (1/2). However, if the maximum number of employees (10) are women, then the probability that both are women is 1. p could be greater or less than one half.

Statement (2) is also insufficient. Clearly, the number of men could be 0 or 1--in either case, the probability of choosing two men is 0. Try a few possibilities--when working the problem multiple times, each time gets faster:

2 men: ((2/10))((1/9)) = (1/45) - this is possible

3 men: ((3/10))((2/9)) = (1/15) - this is possible

4 men: ((4/10))((3/9)) = (2/15) - too big.

Thus, there could be anywhere from 0 to 3 men. We already know that if there are 0 or 1 men, the probability that both are women is far greater than (1/2). Let's try the other extreme. If there are 3 men, there are 7 women, meaning the probability of choosing two women is:

((7/10))((6/9)) = (42/90) = (7/15)

That's slightly less than (1/2), so p could be less than or greater than (1/2).

Taken together, the statements are still insufficient. (2) tells us that there are between 7 and 10 women. (1) is even less specific. Putting them together, we only know that there are between 7 and 10 women, meaning that the probability of choosing two women is between (7/15) and 1. Choice (E) is correct.

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