Sock probability
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When I pull two socks from a pile, they usually match.

I wear black socks with separated toes and turned heels. Because the inner toe is larger than the outer toe and there’s a heel, left and right socks are distinct.

When I launder my socks, I sometimes pair them all up but other times I simply put them in a pile. Taking them from the pile, it seems like more often than not the first two socks I grab are a matched pair. Could that be?

Rather than keeping tally to correct my instinct, I decides to do a bit of math:

• When I’m down to my last pair, I’ll get a matching pair every time.
• When I have two pairs left, I grab a sock and then of the remaining 3 socks, 2 are matches and 1 is not, so a ⅔ chance of a match.
• When I have three pairs left, I grab a sock and then of the remaining 5 socks, 3 are matches and 2 is not, so a ⅗ chance of a match.
• When I have x pairs left, I grab a sock and then of the remaining 2x − 1 socks, x are matches and x − 1 is not, so a  x 2x − 1
chance of a match.

The more socks I have, the closer to 50/50 it gets, but I’m always more likely to get a matching pair than not. As a pull socks from a pile of ten pairs, removing them from the pile, I expect to get a matched pair on the first try just over six of the ten times I put on a pair of socks.