# Meteorological Impossibilities

##### By Bryan Caplan

The Weather Channel’s daily and hourly forecasts often seem logically incompatible. Consider Oakton, VA’s forecast for today. The current daily prediction says “60% chance of rain.” But several evening hours *individually* have the same probability of 60%. Unless I’m missing something, this is only possible if those probabilities are perfectly *dependent* (if rain happens, it happens during every hour) or negatively independent (if rain happens one hour, it *doesn’t* happen during other hours).

These extreme cases seems unlikely. The ironclad puzzle, though, is that the current forecast for 7 PM is a *70%* chance of rain. How can an hour have 70% when the whole day only has 60%? Nor is this a fluke case; in my experience, hourly rain probabilities slightly above the daily probabilities pop up every few days.

I’m tempted to dismiss my own puzzlement by quoting *The Simpsons*:

Comic Book Guy: Last night’sItchy & Scratchy

was, without a doubt, the worst episode ever. Rest assured I was on the

Internet within minutes registering my disgust throughout the world.

Bart: Hey, I know it wasn’t great, but what right do you have to complain?

Comic Book Guy: As a loyal viewer, I feel they owe me.

Bart: For what? They’re giving you thousands of hours of

entertainment for free. What could they possibly owe you? If anything,

you owethem.

Comic Book Guy: …Worst episode ever.

Fair point, but is there anything I’m missing?

__Update:__ Minutes after writing this post, I realized that the problem is more severe than I thought. The daily and multiple hourly forecasts can indeed be equal if the probabilities are perfectly dependent (or nearly perfectly dependent, with a slight rounding error). But the “negative dependence” loophole I suggested is completely confused. If there is a 60% chance at 6 PM and 7 PM, and rain doesn’t happen at 6 PM, then *any* lingering positive probability of rain at 7 PM implies that the probability of rain for the day initially *exceeded* 60%. This is true for partial dependence, independence, and negative dependence.