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?
Instead of figuring out how many socks there are, pretend that there are an infinite number. Or if you don't like that, then sample with replacement (ie, put the first sock back after selecting it). Yes, I know there are practical problems with both of these. In the first case, the volume of the drawer is insufficient. In the second case, you are not holding two socks at the end. But those are engineering concerns, not mathematical ones.
Then, the probability of picking one black sock = sqrt(2/3).
The probability of picking one white sock = 1-sqrt(2/3).
The probability of picking two white socks = (1-sqrt(2/3))^2 =
which is approximately 3.37%
I notice that this quick calculation is close to both the part (a) and (b) answers.
An approximation
