Lansing got some record-breaking rain this week. Tuesday we got over two inches of rain, doubling the hundred-plus-year-old previous record. I mention because it got me to wondering how often we should expect records to break. I mean if the thing being measured probably isn’t changing. So my inspiration is out, as there’s no serious question about the climate changing. Measures of sports performance are also no good.
But we can imagine there’s something with an underlying property that isn’t changing. So if you keep getting samples of some independent, normally-distributed property in, how often should you expect to go between record-setting values? New records should start pretty thick on the ground. The first value is necessarily both a new high and low. The second is either a high or a low. The third seems to have a good chance of being a new extreme. Fourth, too. But somewhere along the way extremes should get rarer. Even if the 10,000th sample recorded is a new record high or low, what are the odds the 10,001st is? The 10,010th?
Haven’t got an answer offhand, although it’s surely available. Just mulling over how to attack the problem before I do what I always do and write a Matlab program to do a bunch of simulations. Easier than thinking. But I’ll leave the problem out for someone needing the challenge.
Every person who ever suffered through that innocent-looking problem where you’re given the size of a record and data about how wide the groove is and asked how many are on the side of the record and then after a lot of confused algebra handed in an answer and discovered it was a trick question has that burned into their brain, and maybe still resents the teacher or book of math puzzles that presented them with the challenge only to have the disappointing answer revealed.
This may be a generational frustration. I think but don’t know that compact discs and DVDs actually have concentric rings so that the how-many-grooves equivalent would be a meaningful, non-trick question; to check would require I make the slightest effort so I’ll just trust that if I’m wrong someone will complain. In another thirty years the word problem may have disappeared from the inventory. But it irritated me, and my Dearly Beloved, and I’m sure irritated other people too. And, yes, we’ve all heard of those novelty records where there’s two or three grooves on a side and you don’t know until fairly well into the performance which version you’re listening to, but I’ve never actually held one in my hand, and neither have you. For the sake of this discussion we may ignore them.
But the question we plunge into answering before we’ve noticed the trick is more like this: If we drew a line from the hole in the center straight out, a radial line if I want to make this sound mathematical, then it crosses some number of grooves; how many? Or maybe like this: how many times does the groove go around the center of the record? And that’s interesting. And I want to describe how I’d work out the problem — in fact, how I did work it out a few nights ago — including a major false start and how that got me to a satisfactory answer.