Probability - 2 Pairs of Socks

This topic has 7 expert replies and 4 member replies
bml1105 Really wants to Beat The GMAT!
Joined
24 Nov 2013
Posted:
113 messages
Followed by:
5 members
Thanked:
1 times
Probability - 2 Pairs of Socks Post Sun May 25, 2014 4:02 pm
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    In packing for a trip, Sarah puts three pairs of socks - one 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?

    (A) 1/5

    (B) 1/4

    (C) 1/3

    (D) 2/3

    (E) 4/5

    OA: A

    Thanked by: moumi2013
    Need free GMAT or MBA advice from an expert? Register for Beat The GMAT now and post your question in these forums!
    Post Sun May 25, 2014 4:31 pm
    bml1105 wrote:
    In packing for a trip, Sarah puts three pairs of socks - one 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?

    (A) 1/5

    (B) 1/4

    (C) 1/3

    (D) 2/3

    (E) 4/5

    OA: A
    Number of ways to choose 4 socks from 6 options = 6C4 = (6*5*4*3)/(4*3*2*1) = 15.
    Number of ways to choose 2 matching pairs = 3. (RRBB, RRGG, BBGG)
    P(selecting 2 matching pairs) = 3/15 = 1/5.

    The correct answer is A.

    _________________
    Mitch Hunt
    GMAT Private Tutor
    GMATGuruNY@gmail.com
    If you find one of my posts helpful, please take a moment to click on the "Thank" icon.
    Available for tutoring in NYC and long-distance.
    For more information, please email me at GMATGuruNY@gmail.com.



    Last edited by GMATGuruNY on Tue May 27, 2014 2:55 am; edited 1 time in total

    Thanked by: moumi2013
    Free GMAT Practice Test How can you improve your test score if you don't know your baseline score? Take a free online practice exam. Get started on achieving your dream score today! Sign up now.
    theCodeToGMAT GMAT Titan
    Joined
    14 Aug 2012
    Posted:
    1556 messages
    Followed by:
    34 members
    Thanked:
    448 times
    Target GMAT Score:
    750
    GMAT Score:
    650
    Post Mon May 26, 2014 1:52 am
    RR, GG, BB ==> 4 SOCKS

    Probability of Two pairs = RRGG + RRBB + GGBB

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

    18 * 1/3 * 1/5 * 1/2 * 1/3

    1/5

    {A}

    _________________
    R A H U L



    Last edited by theCodeToGMAT on Wed May 28, 2014 1:48 am; edited 1 time in total

    Post Mon May 26, 2014 11:07 am
    Hi bml1105,

    In probability questions, there are only 2 things that you can calculate: what you WANT and what you DON'T WANT.

    (WANT) + (DON'T WANT) = 1

    In this question, we can calculate the probability of what we DON'T WANT and subtract it from 1 to figure out the probability of what we do WANT.

    The question asks for the probability of pulling 4 socks out that form 2 matching pairs. For this to occur, the two socks that are left would also form a matching pair. If the 2 leftover socks DO NOT form a matching pair, then the 4 socks that are pulled will NOT form 2 matching pairs.

    Probability of 2 socks NOT forming a matching pair…

    1st sock = 1 (any of the socks can be the first sock)
    2nd sock = 4/5 (since there's only one sock that matches the first sock).

    Probability of NOT forming a pair with 2 socks: = 1 x 4/5 = 4/5 (which ALSO means a 4/5 chance of NOT having 2 matching pairs of 2 socks)

    1 - 4/5 = 1/5 (meaning a 1/5 chance of having 2 matching pairs of 2 socks).

    Final Answer: A

    GMAT assassins aren't born, they're made,
    Rich

    _________________
    Contact Rich at Rich.C@empowergmat.com

    bml1105 Really wants to Beat The GMAT!
    Joined
    24 Nov 2013
    Posted:
    113 messages
    Followed by:
    5 members
    Thanked:
    1 times
    Post Mon May 26, 2014 7:44 pm
    I think where I get confused with this question is that if there are three pairs of socks (6 total socks) and 4 are randomly selected, a pair of socks must be chosen. It doesn't matter which the pair are (red, green or blue), but that leaves one pair out of the loop for being chosen and limits are choices for the last two socks to 4 socks.

    I took that pair out and then tried to do the problem both ways: (1) (probability of one sock pulled and the second sock matching) x (1 pair that was definitely pulled) and (2) (1 - probability of one sock pulled and the second sock not matching) x (1 pair that was definitely pulled). Both ways I got (C) 1/3

    Should it be a general rule that I never take out the pair that is definitely made?

    Post Tue May 27, 2014 7:13 am
    bml1105 wrote:
    In packing for a trip, Sarah puts three pairs of socks - one 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?

    (A) 1/5
    (B) 1/4
    (C) 1/3
    (D) 2/3
    (E) 4/5

    IMPORTANT: As Rich has noted, if there are 2 pairs among the 4 SELECTED socks, then the 2 socks that are NOT SELECTED must also form a pair.
    So, P(2 pairs among the 4 selected socks) = P(2 NOT SELECTED socks are paired)

    P(2 NOT SELECTED socks are paired) = P(1st not selected sock is any color AND 2nd not selected sock matches the 1st)
    = P(1st not selected sock is any color) x P(2nd not selected sock matches the 1st)
    = 1 x 1/5
    = 1/5
    = A

    Cheers,
    Brent

    _________________
    Brent Hanneson – Founder of GMATPrepNow.com
    Use our video course along with Beat The GMAT's free 60-Day Study Guide

    Enter our contest to win a free course.

    GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMAT’s FREE 60-Day Study Guide and reach your target score in 2 months!
    Post Tue May 27, 2014 8:40 am
    bml1105 wrote:
    In packing for a trip, Sarah puts three pairs of socks - one 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?

    (A) 1/5
    (B) 1/4
    (C) 1/3
    (D) 2/3
    (E) 4/5

    We can also use counting methods to solve this question.

    IMPORTANT: If there are 2 pairs among the 4 SELECTED socks, then the 2 socks that are NOT SELECTED must also form a pair.
    So, P(2 pairs among the 4 selected socks) = P(2 NOT SELECTED socks are paired)

    P(2 NOT SELECTED socks are paired) = (# outcomes in which the 2 socks are paired)/(total # of ways to have 2 not selected socks)

    # outcomes in which the 2 socks are paired
    We have have 2 red socks, 2 blue socks or 2 green socks.
    So, there are 3 outcomes in which the 2 socks are paired

    total # of ways to have 2 not selected socks
    There are 6 socks altogether, and we want to not select 2 of them.
    Since the order doesn't matter here, we can use combinations.
    We can choose 2 socks from 6 socks in 6C2 ways (15 ways)

    Aside: If anyone is interested, we have a free video on calculating combinations (like 6C2) in your head: http://www.gmatprepnow.com/module/gmat-counting?id=789

    So, P(2 NOT SELECTED socks are paired) = 3/15
    = 1/5
    = A

    Cheers,
    Brent

    _________________
    Brent Hanneson – Founder of GMATPrepNow.com
    Use our video course along with Beat The GMAT's free 60-Day Study Guide

    Enter our contest to win a free course.

    GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMAT’s FREE 60-Day Study Guide and reach your target score in 2 months!
    Post Tue May 27, 2014 9:07 am
    bml1105 wrote:
    I think where I get confused with this question is that if there are three pairs of socks (6 total socks) and 4 are randomly selected, a pair of socks must be chosen. It doesn't matter which the pair are (red, green or blue), but that leaves one pair out of the loop for being chosen and limits are choices for the last two socks to 4 socks.

    I took that pair out and then tried to do the problem both ways: (1) (probability of one sock pulled and the second sock matching) x (1 pair that was definitely pulled) and (2) (1 - probability of one sock pulled and the second sock not matching) x (1 pair that was definitely pulled). Both ways I got (C) 1/3

    Should it be a general rule that I never take out the pair that is definitely made?
    In taking out a matching pair, you imply that the first 2 socks selected will DEFINITELY form a matching pair.
    Not so.
    In fact, it is far more likely that the first 2 socks selected will NOT form a matching pair.
    Since you take it as a GIVEN that the first 2 socks will form a matching pair, you INCREASE the odds of selecting two matching pairs.
    The result is that your answer (1/3) is greater than the ACTUAL probability of selecting 2 matching pairs (1/5).

    _________________
    Mitch Hunt
    GMAT Private Tutor
    GMATGuruNY@gmail.com
    If you find one of my posts helpful, please take a moment to click on the "Thank" icon.
    Available for tutoring in NYC and long-distance.
    For more information, please email me at GMATGuruNY@gmail.com.

    Thanked by: bml1105
    Free GMAT Practice Test How can you improve your test score if you don't know your baseline score? Take a free online practice exam. Get started on achieving your dream score today! Sign up now.
    Post Tue May 27, 2014 10:56 am
    One more approach:

    The first sock selected can be any color.
    Of the 5 remaining socks, 3 will be selected.
    Thus, the probability that a sock will be selected to match the first = 3/5.
    The third sock selected can be any color.
    Of the 3 remaining socks, 1 will be selected.
    Thus, the probability that the last sock selected will match the third = 1/3.
    To combine these probabilities, we multiply:
    3/5 * 1/3 = 1/5.

    _________________
    Mitch Hunt
    GMAT Private Tutor
    GMATGuruNY@gmail.com
    If you find one of my posts helpful, please take a moment to click on the "Thank" icon.
    Available for tutoring in NYC and long-distance.
    For more information, please email me at GMATGuruNY@gmail.com.

    Thanked by: moron
    Free GMAT Practice Test How can you improve your test score if you don't know your baseline score? Take a free online practice exam. Get started on achieving your dream score today! Sign up now.
    bml1105 Really wants to Beat The GMAT!
    Joined
    24 Nov 2013
    Posted:
    113 messages
    Followed by:
    5 members
    Thanked:
    1 times
    Post Thu May 29, 2014 4:42 pm
    Thank you everyone! It finally clicked.

    Zoser Just gettin' started!
    Joined
    16 Jan 2017
    Posted:
    22 messages
    Post Sat May 27, 2017 10:11 am
    Quote:
    The first sock selected can be any color.
    Of the 5 remaining socks, 3 will be selected.
    Isn't the second selection to match the first color is 1/5 not 3/5?

    Post Sat May 27, 2017 12:15 pm
    Zoser wrote:
    Quote:
    The first sock selected can be any color.
    Of the 5 remaining socks, 3 will be selected.
    Isn't the second selection to match the first color is 1/5 not 3/5?
    For two matching pairs to be selected, the first sock selected must match one of the other 3 socks selected.
    Once the first sock has been selected, 3 of the remaining 5 socks will then be selected.
    Thus, the probability that this group of 3 will provide a match for the first sock is 3/5.

    _________________
    Mitch Hunt
    GMAT Private Tutor
    GMATGuruNY@gmail.com
    If you find one of my posts helpful, please take a moment to click on the "Thank" icon.
    Available for tutoring in NYC and long-distance.
    For more information, please email me at GMATGuruNY@gmail.com.

    Free GMAT Practice Test How can you improve your test score if you don't know your baseline score? Take a free online practice exam. Get started on achieving your dream score today! Sign up now.

    Best Conversation Starters

    1 hazelnut01 59 topics
    2 rsarashi 21 topics
    3 NandishSS 18 topics
    4 richachampion 16 topics
    5 GMATinsight 15 topics
    See More Top Beat The GMAT Members...

    Most Active Experts

    1 image description GMATGuruNY

    The Princeton Review Teacher

    115 posts
    2 image description Rich.C@EMPOWERgma...

    EMPOWERgmat

    99 posts
    3 image description Brent@GMATPrepNow

    GMAT Prep Now Teacher

    91 posts
    4 image description Matt@VeritasPrep

    Veritas Prep

    80 posts
    5 image description DavidG@VeritasPrep

    Veritas Prep

    67 posts
    See More Top Beat The GMAT Experts