# Possibility of getting odd pair socks

My son's drawer has 3 black socks and 2 white socks. He cannot understand that, when eyes closed and tries to get a pair, why in 1st attempt same color pair is never found, it is usually mismatched. what he does if pair is mismatched he throws back in the drawer ! He has asked me this few times.. I cannot answer him ...( we are not color blind!) thank you. Rodney

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1. I think this is not real-analysis. 2. Draw a small tree chart and mark all cases you have and you already solved this. (Consider the probabilitys are not all equal) –  Listing Jun 25 '11 at 11:45
Note that the probability of a match is not dramatically less than $1/2$. Perhaps the "usually mismatched" perception comes from the fact that we remember mismatches more than matches, just like we remember wrong weather forecasts more than right ones. –  André Nicolas Jun 25 '11 at 13:10

There are $$\frac{5 \times 4}{1 \times 2} = 10$$ ways of choosing two socks from a drawer containing five socks. There are three ways of choosing a matching pair of black socks and one way of choosing a matching pair of white socks. Thus there are four ways of choosing a matching pair of socks. The probability of choosing a matching pair of socks is $$\frac{4}{10}.$$

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Jay has already provided a good answer, but it often helps people to see a different approach to a problem.

To pull out a pair of black socks, you have to pick a black sock on the first pull (probability 3/5), and then pull out a black sock on the second pull (probability 2/4), giving overall a 6/20 probability of pulling a pair of black socks.

To pull a pair of white socks, you must first pull one (probability 2/5) then pull the remaining one (probability 1/4), giving an overall probability of 2/20.

The probability of pulling a black pair or a white pair is the sum of those, or 6/20 + 2/20 = 8/20, which is the same thing Jay got combinatorically.

BTW, the way you should pull socks from the drawer is to pull two socks. If they are matched, great! If they are not matched, pull a third sock. You will then have a matching pair, and an odd sock that you can throw back in the drawer. You'll end up wearing black socks 7/10 of the time and white socks 3/10 of the time.

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