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?

Let's just say that there are x socks and y of them are black.  The chance of taking out a black pair (implying without replacement) is the following equation:

y*(y-1)
x*(x-1)

More important then solving this equation is noting that the basic form: the product of two consecutive integers divided by the product of two more consecutive integers.

What I did was write a list of possible products of consecutive integers.  Starting with 1*2, this is it.

2
6
12
20
30
42
56
72
90
110
132
156
182
210
240
272
306
342
380
420
462

This is certainly not long enough for part 2 (because I did it by hand) but it shows the immediate solution of 6*5/5*4 (or 20/30).  So the simplest solution is that there are 6 socks, 5 of which are black.  In this case the probability of drawing a white pair is 0.

For part 2, it would seem to me to be impractical to use the same method.

Posted by Tristan on 2004-09-01 17:45:48