How Often Should Records Break? A Puzzle For You

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.