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Pairs of Socks (Posted on 2004-09-01) Difficulty: 3 of 5
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?

Bonus: what would the answer be if the drawer contained between 100 and 1000 socks?

See The Solution Submitted by SilverKnight    
Rating: 3.0000 (1 votes)

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Solution solution | Comment 9 of 20 |

If there are b black socks and w white socks

b(b-1)/((b+w)(b+w-1)) = 2/3

Which leads to the quadratic

b^2 - (1+4w)b -2w^2+2w = 0

When the quadratic formula is used to find b in terms of w, the positive integral solutions come out for the following values of w (up to how far I took it):

  w       b
1       5
10 45
99 441
980 4361

The first solution doesn't have at least one pair of white socks.

The second fits part 1, and gives a probability of drawing a white pair as 10*9/(55*54) = .030303... = 1/33.

The third fits part 2, and gives a probability of drawing a white pair as 99*98/(540*539)=.033333... = 1/30.

The fourth gives a probability of 980*979/(5341*5340)=0.033639143730887... =11/327 = 1/29.727272...

As the number of socks approaches infinity, this resembles the probability of drawing homozygous alleles in a gene pool in Hardy-Weinberg equilibrium. The probability would be the same as if drawing were with replacement. The proportion of black socks (or genes) would be sqrt(2/3) or about 0.816496580927726, leaving 0.183503419072274 white, for a probability of 0.033673504811215 of drawing a pair of white socks or genes (1/29.69693845669907).

Calculations of the solutions to the quadratic were done by entering a formula into a Reverse Polish Notation calculator program that tabularized the results.  The RPN is:

dup, dup, 4 *, 1+, x^2, 8 *, swap, 8 *, - + sqrt, swap, 4 * +, 1+2/

corrected transcription of the first equation

Edited on September 3, 2004, 9:31 am
  Posted by Charlie on 2004-09-01 20:03:59

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