You are in a pitch black room and need to get a pair of socks out of your drawer which can contain up to 100 socks. In the drawer is a mixture of black and white socks, and there's at least one pair of either color. If you choose two socks, the chance that you draw out a black pair is 2/3.

What is the chance that you draw out a white pair?
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Bonus: what would the answer be if the drawer contained between 100 and 1000 socks?

The probability of drawing a black pair is the probability of drawing a black sock squared. If we take that relationship in reverse, the proportion of black socks in the drawer is the square root of the probability of drawing a pair:

sqrt (2/3) = (around) 0.82

assuming a sock is either one color or another the proportion of white socks is 0.18. The probability of drawing a pair is:

(0.18)^2 = (around) .033 which is about 1 in 30.

the solution to the second part is identical, except more significant digists come into play. The proportion of black socks can be taken out to 0.816 making the proportion of white socks .184.

The rounding in this solution may seem inexact, but that's because there is some rounding inherent in the problem. You can't have exactly a 2/3 chance of drawing a black pair out of 100 socks.

Posted by Nate on 2004-09-03 15:30:09