April 2012 Brain Teaser Solution

Q:You are in a pitch dark room selecting socks from a drawer that has only six socks, a mixture of black and white. If you choose two socks, the chances that you draw out a white pair is 2/3. What are the chances that you draw out a black pair?


A: 0 (it is impossible)


The good wrong answer is 1/3 because that is the remainder of the probability. This assumes that the pair of black socks has the same probability as a pair of white socks but there is also a chance that you pull out one black and one white.  The combinations with 4 white (W1, W2, W3 and W4) and 2 black (B1 and B2) are: 

W1W2, W1W3, W1W4, W2W3, W2W4, W3W4, W1B1, W2B1, W3B1, W4B1, W1B2, W2B2, W3B2, W4B2, and (last but not least) B1B2.  Far less than two-thirds of the aforementioned are a white pair.


So let's try 5 white and 1 black:  W1W2, W1W3, W1W4, W1W5, W2W3, W2W4, W2W5, W3W4, W3W5, W4W5, W1B1, W2B1, W3B1, W4B1, W5B1makes a total of 15 combination of which 10 are white pairs.  This simplifies to 2/3 so there must be 5 white and 1 black.  But with only one black sock, you can never get a black PAIR!