Company X employs more than 5 people, and will choose 2 employees at

Company X employs more than 5 people, and will choose 2 employees at random to send to a business meeting. Is the probability that Company X will select 2 women greater than 0.5?
Let number of male employees = M
Number of female employees = W
Now, M + W > 5
Probability of selecting two women for the meeting \(\frac{W}{M+W }* \frac{W-1}{M+W-1}\) > 0.5 ??

(1) More than 70% of the employees of Company X are women.
Case I: M = 1, W = 5
\(Prob. = \frac{W}{M+W }* \frac{W-1}{M+W-1} = \frac{5}{6} * \frac{4}{5} = \frac{2}{3}\) YES

Case II: M = 2, W = 5
\(Prob. = \frac{W}{M+W }* \frac{W-1}{M+W-1} = \frac{5}{7} * \frac{4}{6} = \frac{10}{21}\) NO

INSUFFICIENT.

(2) Company X employs more than 12 people in total.
M + W > 12
Case I: M = 1, W = 12
\(Prob. = \frac{W}{M+W }* \frac{W-1}{M+W-1} = \frac{12}{13} * \frac{11}{12} = \frac{11}{13}\) YES

Case II: M = 6, W = 6
\(Prob. = \frac{W}{M+W }* \frac{W-1}{M+W-1} = \frac{6}{12} * \frac{5}{11} = \frac{5}{22}\) NO

INSUFFICIENT.

Together 1 and 2
M + W > 12 and W > 0.7*(M + W) OR
\(\frac{10}{7}\)W > M + W

Adding both
M + W + \(\frac{10}{7}\)W > 12 + M + W
W > \(\frac{84}{10}\)
Thus, W ≥ 9

But this does not satisfies the condition of more than 70% women So,
W ≥ 10

Now,
Case I: M = 3, W = 10
\(Prob. = \frac{W}{M+W }* \frac{W-1}{M+W-1} = \frac{10}{13} * \frac{9}{12} = \frac{15}{26}\) YES
Till this point any value of W > 10 would result in probability of selecting 2 women for meeting be more than 0.5.

BUT

Many cases are possible e.g. if W = 11 and M = 20
\(Prob. = \frac{W}{M+W }* \frac{W-1}{M+W-1} = \frac{11}{20} * \frac{10}{19} = \frac{11}{38}\) NO

HENCE,
INSUFFICIENT.

ANSWER E.

We are to determine whether the probability of selecting two females to represent the company is more than 0.5.

Statement 1 says that the company has more than 70% of its employees as female.

If there are a total of 1000 employees, implying there could be 701 are female while 299 are male, then
P(2W) = 701/1000 * 700/999 < 0.5
However, if there are 900 females and 100 males, then
P(2W) = 900*899/(1000*999) > 0.5

Statement 1 is insufficient.

Statement 2: Company X employs more than 12 people in total.
Statement 2 is clearly insufficient. We don't know the ratio of females to males, hence statement 2 is not enough to determine the probability of selecting 2 females to represent the company.

1+2
Still insufficient. Since a total of 1000 employees is more than 12, and this option leads to yes and no in statement 1, hence both statements when combined are not sufficient.

The answer is E.

Call the probability that Company X will select 2 women = P

Statement (1): More than 70% of the employees of Company X are women.
P sometimes is less than 0.5, sometimes is greater than 0.5, depending on the rate of women employees in the company and the total of employs.
=> Not suff

Statement (2): Company X employs more than 12 people in total.
Don't know how many employs are women or the ratio of women compared with the total of employs of Company X.
=> Not suff

Combine: (1) & (2)
Sure P>0.5 whatever the rate of women employees in the company more than 70% and the total of employs more than 12 people.

=>> Choice C

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.