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## Matching Socks

Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

 Puzzle ID: #14164 Fun: (2.76) Difficulty: (2.13) Category: Probability Submitted By: mathemagician

Mismatched Joe is in a pitch dark room selecting socks from his drawer. He has only six socks in his drawer, a mixture of black and white. If he chooses two socks, the chances that he draws out a white pair is 2/3. What are the chances that he draws out a black pair?

He has a ZERO chance of drawing out a black pair.

Since there is a 2/3 chance of drawing a white pair, then there MUST be 5 white socks and only 1 black sock. The chances of drawing two whites would thus be: 5/6 x 4/5 = 2/3 . With only 1 black sock, there is no chance of drawing a black pair.

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 jacintan Aug 29, 2003 COOL! jacintan Aug 29, 2003 COOL! jamesernst Mar 03, 2004 Actually, your answer is wrong. 45 White socks and 10 black socks also allows for a 2/3 chance of drawing a white pair. I am sure there are others. ghost09 Apr 11, 2004 WRONG ANSWER Poker Aug 01, 2004 That may be, jamesernst, but he said six socks. That leaves only the 5-1 distribution. tsimkin Nov 23, 2004 Excellent question (and it is the right answer.) brianz May 16, 2005 Good one! Really fun. Jimbo Mar 05, 2006 On the other hand he could have had -4 white socks and 10 black socks in his drawer. Probability of drawing a white sock = -4/6. That would now make -5 white socks in the drawer. Probability of drawing another white sock is -5/5. Therefore the probability of drawing 2 white socks = (-4/6)x(-5/5) = 2/3. See it works! Then the probability of drawing 2 black socks is (10/6)x(9/5) = 3 Doh! I think if you draw out a negative white sock, the number of socks in the drawer increases Seriously good teaser even if I did start solving huge equations to get it! calmsavior Nov 19, 2006 Jimbo, try being a little more ... realistic? donga Aug 19, 2008 i think the question is wrong i have seen a similar question else where.., a pair of socks consists of a left sock and a right sock. (if i'm right) u can never have a prob of 2/3 of picking a pair of white socks from six socks donga Aug 25, 2008 sry for the above comment.. just thot there will be two kinds of sox... opqpop Jan 26, 2010 Took me about 2 minutes because I started thinking about combinations like 6 choose 2 is 15, so out of these 10 has to be white pairs, and then became stumped for a bit. Then I realized probably of drawing 2 whites is just x/6 * (x-1)/5 with x being the number of white socks. Sigh I wish I could get the insight to just get to this step in the first place -_- opqpop Sep 11, 2010 Haha the me 7 months ago sure was dumb. You can do this easily w/ combinations as well. The key is to note the number of white socks choose 2 must equal 10 on the numerator. Hence, there are 5 white socks. Jayo Apr 27, 2011 Loved it!!!! Didn't solve it though