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# A 4-person committee is to be formed from a group of 6 Democrats and

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Joined: 25 Sep 2018
Posts: 872
Location: United States (CA)
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A 4-person committee is to be formed from a group of 6 Democrats and  [#permalink]

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22 Apr 2019, 21:46
00:00

Difficulty:

85% (hard)

Question Stats:

47% (02:28) correct 53% (03:01) wrong based on 32 sessions

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A 4-person committee is to be formed from a group of 6 Democrats and 5 Republicans. Which of the following is the closest approximation to the probability that the committee will consist of at least 3 Democrats?
A)0.11
B)0.22
C)0.35
D)0.42
E) 0.75

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Posts: 3410
Re: A 4-person committee is to be formed from a group of 6 Democrats and  [#permalink]

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22 Apr 2019, 23:13

Solution

Given:
• A 4-person committee is to be formed from a group of 6 Democrats and 5 Republicans

To find:
• The probability (closest value) that the committee will consist of at least 3 Democrats

Approach and Working:
• Number of ways the committee can be formed with at least 3 Democrats = 3D AND 1R OR 4D = $$^6C_3 * ^5C_1 + ^6C_4 = 20 * 5 + 15 = 115$$
• Total number of ways a 4-member committee can be formed = $$^{11}C_4 = 330$$
• Therefore, the probability = $$\frac{115}{330} = 0.35$$

Hence, the correct answer is option C.

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Re: A 4-person committee is to be formed from a group of 6 Democrats and  [#permalink]

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25 Apr 2019, 11:07
Abhi077 wrote:
A 4-person committee is to be formed from a group of 6 Democrats and 5 Republicans. Which of the following is the closest approximation to the probability that the committee will consist of at least 3 Democrats?
A)0.11
B)0.22
C)0.35
D)0.42
E) 0.75

Can someone solve this using the Probability method, instead of combinations?
_________________
"Luck is when preparation meets opportunity!"
Manager
Joined: 28 Jan 2019
Posts: 96
Location: India
GMAT 1: 700 Q49 V36
GPA: 4
WE: Manufacturing and Production (Manufacturing)
Re: A 4-person committee is to be formed from a group of 6 Democrats and  [#permalink]

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26 Apr 2019, 07:08
Abhi077 wrote:
A 4-person committee is to be formed from a group of 6 Democrats and 5 Republicans. Which of the following is the closest approximation to the probability that the committee will consist of at least 3 Democrats?
A)0.11
B)0.22
C)0.35
D)0.42
E) 0.75

Ok, here's how I did it.

Atleast 3 Democrats = P(3D, 1R) + P(4D)

P(3D, 1R) = $$\frac{6}{11}*\frac{5}{10}*\frac{4}{9}*\frac{5}{8}$$ = $$\frac{5}{66}$$
Now, there are 4 ways to get this possibility = $$4*\frac{5}{66} = \frac{20}{66}$$

P(4D) = $$\frac{6}{11}*\frac{5}{10}*\frac{4}{9}*\frac{3}{8}$$ = $$\frac{1}{22}$$
Only 1 way to get this possibility.

Total probability P = $$\frac{20}{66}+\frac{1}{22} = \frac{23}{66}$$
Now, $$\frac{22}{66}=0.33$$
Therefore, $$\frac{23}{66}$$ ~= 0.35.
_________________
"Luck is when preparation meets opportunity!"
Re: A 4-person committee is to be formed from a group of 6 Democrats and   [#permalink] 26 Apr 2019, 07:08