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’s Itchy & 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 owe them.
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.