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## Official Guide Explanation:

Data Sufficiency #122

**Background**

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**Solution and Metadata**

**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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