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Old Forum Question With Slight Changes

by moumi2013 » Tue May 27, 2014 9:44 pm
I have slightly changed another Forum question at https://www.beatthegmat.com/probability- ... 76629.html to generate this question:

In packing for a trip, Sarah puts four pairs of socks - two red, one blue, and one green - into one compartment of her suitcase. If she then pulls four individual socks out of the suitcase, simultaneously and at random, what is the probability that she pulls out exactly two matching pairs?
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by theCodeToGMAT » Wed May 28, 2014 1:46 am
According to me,
Red = 4
Blue = 2
Green = 2

RRRR --> 1 way

4/8 * 3/7 * 2/6 * 1/5

RRBB --> 4!/2!2! = 6 ways

RRGG --> 4!/2!2! = 6 ways
Ways = 4/8 * 3/7 * 2/6 * 1/5

BBGG --> 4!/2!2! = 6 ways
Ways = 2/8 * 1/7 * 2/6 * 1/5

Total ways = 13 * 4/8 * 3/7 * 2/6 * 1/5 + 6 * 2/8 * 1/7 * 2/6 * 1/5

= 13 * 1/2 * 1/7 * 1/1 * 1/5 + 6 * 1/4 * 1/7 * 1/3 * 1/5

= 13/70 + 1/70 = 1/5
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by GMATGuruNY » Wed May 28, 2014 11:04 am
moumi2013 wrote:I have slightly changed another Forum question at https://www.beatthegmat.com/probability- ... 76629.html to generate this question:

In packing for a trip, Sarah puts four pairs of socks - two red, one blue, and one green - into one compartment of her suitcase. If she then pulls four individual socks out of the suitcase, simultaneously and at random, what is the probability that she pulls out exactly two matching pairs?
All possible cases:
Number of ways to choose 4 socks from 8 options = 8C4 = (8*7*6*5)/(4*3*2*1) = 70.

Good case 1: 4 red are selected
Number of ways = 1. (RRRR.)

Good case 2: 2 red and a non-red matching pair are selected
Number of ways to choose 2 red socks from 4 options = 4C2 = (4*3)/(2*1) = 6.
Number of options for the non-red matching pair = 2. (BB or GG.)
To combine these options, we multiply:
6*2 = 12.

Good case 3: 2 non-red matching pairs are selected
Number of ways = 1. (BBGG.)

(Good cases)/(All possible cases) = (1+12+1)/70 = 14/70 = [spoiler]1/5[/spoiler].
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by moumi2013 » Thu May 29, 2014 2:59 am
was confused on this part: 2 red and a non-red matching pair

(Number of ways to choose 2 red socks from 4 options) * (Number of options for the non-red matching) - CORRECT
OR
(Number of ways to choose 2 red socks from 4 options) + (Number of options for the non-red matching)

Thanks theCodeToGMAT & GMATGuruNY !!

Have got the answers.
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by feedrom » Thu May 29, 2014 10:51 am
GMATGuruNY wrote:
Good case 3: 2 non-red matching pairs are selected
Number of ways = 1. (BBGG.)
Hi Mitch,
Could you please explain:

- Why the position of each individual sock doesn't matter?

- What is the difference between choosing pairs and choosing individuals?
(I am little bit confused with choosing pairs.)

Thank you so much,
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by GMATGuruNY » Thu May 29, 2014 12:30 pm
feedrom wrote:
GMATGuruNY wrote:
Good case 3: 2 non-red matching pairs are selected
Number of ways = 1. (BBGG.)
Hi Mitch,
Could you please explain:

- Why the position of each individual sock doesn't matter?

- What is the difference between choosing pairs and choosing individuals?
(I am little bit confused with choosing pairs.)

Thank you so much,
Problem rephrased:
Yesterday Sue put four pairs of socks - two red, one blue, and one green - into one compartment of her suitcase. Now she holds in her hand 4 socks that she has just removed from this compartment. What is the probability that the 4 socks in Sue's hand are composed of two matching pairs?

The order in which the socks are pulled out doesn't matter.
What matters is whether the COMBINATION OF SOCKS in Sue's hand is composed of two matching pairs.

Let the 8 socks be R�R₂R₃R₄B�B₂G�G₂.

All possible combinations:
From 8 socks, the number of combinations of 4 that can be formed = 8C4 = (8*7*6*5)/(4*3*2*1) = 70.

Good combination 1: 2 pairs of red socks
Only 1 combination is viable:
R�R₂R₃R₄.

Good combination 2: A combination composed of a pair of red socks and a matching pair of non-red socks
The following pairs of red socks may be chosen:
R�R₂
R�R₃
R�R₄
R₂R₃
Râ‚‚Râ‚„
R₃R₄
Total ways = 6.

The following pairs of non-red socks may be chosen:
B�B₂
G�G₂
Total ways = 2.

To combine our options for the red pair with our options for the non-red pair, we multiply:
6*2 = 12.

Good combination 3: 2 matching pairs of non-red socks
Only 1 combination is viable:
B�B₂G�G₂.

Resulting probability:
(good combinations)/(all possible combinations) = (1+12+1)/70 = 14/70 = 1/5.
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