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Probability/ Combinatorics!

by Uva@90 » Fri Oct 18, 2013 7:40 pm
If 2 different representatives are to be selected at random from a group of 10 employees
and if p is the probability that both representatives selected will be women, is p > 1/2

(1) More than 1/2 of the 10 employees are women.
(2) The probability that both representatives selected will be men is less than 1/10

OA is E


Here is my understanding from the question,
Total number of Employees =10. out of which 2 women is to selected
To find:
P(Both Women selcted) >1/2 or not

I am rephrasing the question that Number of women is greater than 5 or not(1/2)
or
Number of women is greater than men or not ?

So from Statement 1 : Number of women is greater than men
So I concluded that A is sufficient

But OA is E

Please help me where I went wrong.

Thanks in advance.

Regards,
uva.
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Source: — Data Sufficiency |
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by theCodeToGMAT » Fri Oct 18, 2013 11:10 pm
For Statement 1:

Women's count = 6, or 7 or 8 or 9 or 10

If "6" Women
p = 6/10 * 5/9 = .6 * ~.5 = ~.3 NO

if "9" Women
p = 9/10 * 8/9 = .9 * ~.8 = ~.7 YES
INSUFFICIENT
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by theCodeToGMAT » Fri Oct 18, 2013 11:16 pm
Aside, Complete Solution:

p - probability of women.

TO find: p>1/2

Statement 1:
Women's count = 6, or 7 or 8 or 9 or 10

If "6" Women
p = 6/10 * 5/9 = .6 * ~.5 = ~.3 NO

if "9" Women
p = 9/10 * 8/9 = .9 * ~.8 = ~.7 YES
INSUFFICIENT

Statement 2:
Men's probability < 1/10
M/10* (M-1)/10 < 1/10

M(M-1) < 10

M can be 0, or 1 or 2, or 3

So, if M=3 W=7
P=7/10*6/9 = .7*~.6 = ~.4 NO

if M=1 W=9
p = 9/10 * 8/9 = .9 * ~.8 = ~.7 YES
INSUFFICIENT

Combining...
Women's Count range from 7 to 10
INSUFFICIENT

Answer [spoiler]{E}[/spoiler]
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by GMATGuruNY » Sat Oct 19, 2013 3:23 am
If 2 different representatives are to be selected at random from a group of 10 employees and if p is the probability that both representatives selected will be women, is p > 1/2 ?

(1) More than 1/2 of the 10 employees are women.
(2) The probability that both representatives selected will be men is less than 1/10.
The following cases satisfy both statements.

7 women, 3 men:
Statement 1: more than 1/2 the employees are women.
Statement 2: P(MM) = 3/10 * 2/9 = 1/15, which is less than 1/10.
Here, P(WW) = 7/10 * 6/9 = 7/15, so p<1/2.

10 women, 0 men:
Statement 1: more than 1/2 the employees are women.
Statement 2: P(MM) = 0.
Here, P(WW) = 1, so p>1/2.

Since p< 1/2 in the first case and p>1/2 in the second case, the two statements combined are INSUFFICIENT.

The correct answer is E.
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