Sock Probability

I don’t know if problems like the following are famous, but there sure are a lot of them online — Cut the Knot, Stack Exchange, and Braingle, for example — and they’re typical for a high school classroom or middle school math competition:

There are 14 red, 6 orange, 10 yellow, 8 green, 4 blue, 12 indigo, and 2 violet socks in my sock drawer. How many socks must I randomly remove from the drawer to guarantee that I have two socks of the same color?

You may or may not know the answer, but the problem itself leads to a follow-up question:

Why the hell do I own socks in every color of the rainbow?

That’s just weird. But if you can get past that, here is a related problem:

If I randomly remove two socks from the drawer, what is the probability that they form a matching pair?

As it turns out, I’m something of a sock aficionado. (Yeah, it’s weird, but surely not surprising. I mean, I write a math jokes blog. You didn’t think I was normal, did you?) Although the context above is fictitious, I am indeed the owner of three pairs of identical socks that look like those shown in the picture. And yes, those are my feet and ankles. My mathematical sexy runs all the way down to my toes.

Here’s a close-up of one of them, in case you can’t see it in the larger picture:

That’s a letter R, because these socks are specially designed for each foot. The other sock has a letter L. (Duh.)

This leads to another mathematical question, more real-world than those above:

I just finished washing these three pairs of socks. While folding them, I selected two socks at random and rolled them together. What’s the probability that there’s one R and one L?

The answer, of course, is zero.

Yes, I know that theoretically the answer should be 3/5. But theory doesn’t match practice in this case. When I do my laundry, I sometimes forget to pay attention to the R and the L, and my sock drawer invariably results in one pair of two R’s, one pair of two L’s, and one correctly matched pair. And then when I wake up at 5:30 a.m. and put on my socks in the dark (so as not to rouse my wife from slumber), my feet feel all weird. The one with the wrong sock starts tingling, so I have to remove the socks and choose another pair entirely.

Similarly, here’s another real-world problem, based on my sock experience:

If there are 10 socks in a load of laundry that I place in the washer and then transfer to the dryer, how many socks will remain when the load is finished drying?

Nine. Yes, I know it’s a cliche. Everyone makes jokes about losing socks. It’s so overdone that the National Comedian’s Guild has declared a moratorium against them. But, I’m not joking. I can’t remember the last time I did a load of laundry and wasn’t missing a sock. I now have a drawer filled with unmatched socks, each like Tiger Woods longing for the return of its Lindsey Vonn.

Sadly, this post is going public just a little too late. Lost Socks Memorial Day was May 9, so we just missed that one. Likewise, we missed No Socks Day on May 8. But there are other holidays in the coming months when you can celebrate the amazing undergarments that protect our feet from our shoes:

• July (exact date TBD): Red Socks Day (commemorating Sir Peter Blake)
• October 4: Odd Socks Day (Australia)
• January (every Friday): Snow Sock Day

And though not an official holiday, there are unlimited Crazy Sock Days happening at elementary, middle, and high schools near you.

99% of socks are single, and you don’t see them crying about it.

How do engineers make a bold fashion statement?
They wear their dark grey socks instead of the light grey ones.

Somewhere, all of my socks, Tupperware lids, and ball point pens are hanging out together, just laughing at me.

—

Because I know you won’t be able to sleep tonight…

• I need to remove 8 socks from my sock drawer to guarantee a color match.
• I don’t actually own socks in every color of the rainbow. Just most colors.
• The probability of selecting two socks from my drawer and getting a matching pair is 23/140.