Anaira Mitch wrote:From a group of 10 people, a committee of four people is to randomly formed. Is the probability greater than 50% that the committee will contain more women than men?
(1) The group from which the committee is formed contains more women than men
(2) The ratio of women to men in the group from which the committee is formed is greater than or equal to 3:2.
Source: Veritas.
(1) The group from which the committee is formed contains more women than men.
Case 1: There are 10 women and no men in the group.
Thus, the probability greater than 50% that the committee will contain more women than men
> 50%. The answer is Yes.
Case 2: There are 6 women and 4 men in the group.
Probability of choosing 4 people such that there are 3 women and 1 man = (6C3 * 4C1) / 10C4 = [(6.5.3)*(4) / (10.9.8.7)] * [(1.2.3.4) / (1.2.3)] = 8/21
Probability of choosing 4 people such that there are 4 women and no man = (6C4) / 10C4 = [(6.5.3.2) / (10.9.8.7)] * [(1.2.3.4) / (1.2.3.4)] = 1/14
Probability of choosing 4 people such that there are more women than man = 8/21 + 1/14 = 19/42
< 1/2. The answer is No.
Insufficient.
(2) The ratio of women to men in the group from which the committee is formed is greater than or equal to 3:2.
Thus, women to men ratio can be 6 : 4, 7 : 3, 4 : 1, or 9 : 1.
Understanding from Statement 1, we get that if there are 6 women and 4 men, the answer is No (Case 2) of Statement 1.
We can deduce that if there are 9 women and 1 man, the answer is Yes. No unique answer. Insufficient.
(1) and (2) combined:
Even after combing the two statements will not help as cases discussed in Statement 2 are applicable here too. Insufficient.
The correct answer:
E
Hope this helps!
-Jay
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