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805+ (Hard)|   Probability|      
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From a group of 10 people, a committee of four people is to 20 Nov 2013, 03:12
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E
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41% (02:34) correct 59% (02:15) wrong based on 514 sessions

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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
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E
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Bunuel
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From a group of 10 people, a committee of four people is to 20 Nov 2013, 03:42
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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?

More women than men in 4-person committee means one of the following 2 cases:
WWWW
WWWM

(1) The group from which the committee is formed contains more women than men.

If there are 10 women in the group then the probability of getting more women will obviously be greater than 50% (100%). I
If there are 6 women and 4 men, then \(P(W>M)=\frac{C^4_6}{C^4_{10}}+\frac{C^3_6*C^1_4}{C^4_{10}}=\frac{15}{210}+\frac{20*4}{210}=\frac{95}{210}<\frac{1}{2}\).

Not sufficient.

(2) The ratio of women to men in the group from which the committee is formed is greater than or equal to 3:2 --> \(\frac{W}{M}\geq{\frac{3}{2}}\). We can get two different answers if we consider 9 women and 1 men and 6 women and 4 men. Not sufficient.

(1)+(2) Consider the same examples as above. Not sufficient.

Answer: E.
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From a group of 10 people, a committee of four people is to 20 Nov 2013, 03:45
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Bunuel
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?

More women than men in 4-person committee means one of the following 2 cases:
WWWW
WWWM

(1) The group from which the committee is formed contains more women than men.

If there are 10 women in the group then the probability of getting more women will obviously be greater than 50% (100%). I
If there are 6 women and 4 men, then \(P(W>M)=\frac{C^4_6}{C^4_{10}}+\frac{C^3_6*C^1_4}{C^4_{10}}=\frac{15}{210}+\frac{20*4}{210}=\frac{95}{210}<\frac{1}{2}\).

Not sufficient.

(2) The ratio of women to men in the group from which the committee is formed is greater than or equal to 3:2 --> \(\frac{W}{M}\geq{\frac{3}{2}}\). We can get two different answers if we consider 9 women and 1 men and 6 women and 4 men. Not sufficient.

(1)+(2) Consider the same examples as above. Not sufficient.

Answer: E.
Show more

Similar questions to practice:
a-committee-consists-of-n-women-and-k-men-in-addition-there-127620.html (700+)
if-2-different-representatives-are-to-be-selected-at-random-128233.html (GMAT Prep 700)
a-committee-of-2-people-is-to-be-formed-from-a-group-of-108096.html (700)
a-certain-panel-is-to-be-composed-of-exactly-three-women-and-108964.html

Hope this helps.
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From a group of 10 people, a committee of four people is to 16 Jan 2014, 20:12
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Bunuel
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?

More women than men in 4-person committee means one of the following 2 cases:
WWWW
WWWM

(1) The group from which the committee is formed contains more women than men.

If there are 10 women in the group then the probability of getting more women will obviously be greater than 50% (100%). I
If there are 6 women and 4 men, then \(P(W>M)=\frac{C^4_6}{C^4_{10}}+\frac{C^3_6*C^1_4}{C^4_{10}}=\frac{15}{210}+\frac{20*4}{210}=\frac{95}{210}<\frac{1}{2}\).

Not sufficient.

(2) The ratio of women to men in the group from which the committee is formed is greater than or equal to 3:2 --> \(\frac{W}{M}\geq{\frac{3}{2}}\). We can get two different answers if we consider 9 women and 1 men and 6 women and 4 men. Not sufficient.

(1)+(2) Consider the same examples as above. Not sufficient.

Answer: E.
Show more

Bunuel,

I am not clear that we you added 6C4 + 6C3x 4C1 in statement 1
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From a group of 10 people, a committee of four people is to 17 Jan 2014, 00:48
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Bunuel
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?

More women than men in 4-person committee means one of the following 2 cases:
WWWW
WWWM

(1) The group from which the committee is formed contains more women than men.

If there are 10 women in the group then the probability of getting more women will obviously be greater than 50% (100%). I
If there are 6 women and 4 men, then \(P(W>M)=\frac{C^4_6}{C^4_{10}}+\frac{C^3_6*C^1_4}{C^4_{10}}=\frac{15}{210}+\frac{20*4}{210}=\frac{95}{210}<\frac{1}{2}\).

Not sufficient.

(2) The ratio of women to men in the group from which the committee is formed is greater than or equal to 3:2 --> \(\frac{W}{M}\geq{\frac{3}{2}}\). We can get two different answers if we consider 9 women and 1 men and 6 women and 4 men. Not sufficient.

(1)+(2) Consider the same examples as above. Not sufficient.

Answer: E.

Bunuel,

I am not clear that we you added 6C4 + 6C3x 4C1 in statement 1
Show more

P(W>M) = P(WWWW) + P(WWWM).

If there are 6 women and 4 men, then, then \(P(WWWW)=\frac{C^4_6}{C^4_{10}}\) and \(P(WWWM)=\frac{C^3_6*C^1_4}{C^4_{10}}\).

Hope it's clear.
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From a group of 10 people, a committee of four people is to 20 Nov 2017, 11:33
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Bunuel
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?

More women than men in 4-person committee means one of the following 2 cases:
WWWW
WWWM

(1) The group from which the committee is formed contains more women than men.

If there are 10 women in the group then the probability of getting more women will obviously be greater than 50% (100%). I
If there are 6 women and 4 men, then \(P(W>M)=\frac{C^4_6}{C^4_{10}}+\frac{C^3_6*C^1_4}{C^4_{10}}=\frac{15}{210}+\frac{20*4}{210}=\frac{95}{210}<\frac{1}{2}\).

Not sufficient.

(2) The ratio of women to men in the group from which the committee is formed is greater than or equal to 3:2 --> \(\frac{W}{M}\geq{\frac{3}{2}}\). We can get two different answers if we consider 9 women and 1 men and 6 women and 4 men. Not sufficient.

(1)+(2) Consider the same examples as above. Not sufficient.

Answer: E.
Show more

Hi Bunuel,

I have one question, are n't both statements saying same thing ?

Statement 1: more women than men => w >= 6 and m <= 4 =>
if w = 6, m = 4, ratio w/m = 3/2
if w = 7, m = 3, ratio w/m = 7/3
ratio w/m >= 3/2

Statement 2: is exactly saying w/m >= 3/2

but answer is definitely (E) as we will get diff probabilities with cases w = 6, m = 4 and w = 7, m = 3 .....

Please clarify

Thanks
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From a group of 10 people, a committee of four people is to 16 Feb 2020, 05:26
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Quick and easy solution without any calculation.

If you have 3 women in the committee of 4 people, ratio of women to men is 3:1.

So, in order to have the probability of more women than men in the committee greater than 0.5, total number of women must be greater than 3/4, i.e. 7.5 out of 10. So, we need at least 8 women in the total population to get the probability higher than 0.5.


Statement-1:
The number of women is higher than number men in the total population.
No. of women > 5.
No. of women may or may not be greater than or equal to 8.

So, statement 1 is not sufficient.

Statement-2:
The number of women is 6 or higher in the total population.
No. of women may or may not be greater than or equal to 8.

So, statement 2 is not sufficient.

Combining both statements:
Even after combining both statements, no. of women may or may not be greater than or equal to 8.

So, ans is E.
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From a group of 10 people, a committee of four people is to 06 Apr 2025, 08:04
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