I used to tell my students that a weather forecast was pretty simple. If it said “40% chance of rain tomorrow” that meant if ten of them went outside, four of them would get wet. They were usually skeptical of this interpretation, to which I give them a lot of credit. After all, they would hear stuff from teachers all day long, and of the stuff they actually listened to, much of it was bullshit. That’s just the nature of schooling: a lot of bullshit gets spread around. Humans are a bullshitting species—we can’t help ourselves.

But it always got me thinking about what such a weather forecast actually meant. Statements of probability are attractive because they are unambiguous. Or ought to be. “It might rain tomorrow” is not very useful. “There’s a really good chance it will rain tomorrow” is a little better.

People may prefer “it will” or “it will not” rain, but hardly anyone is ever that certain. Besides, life is unpredictable. We know this. Probabilities are the best we can do.

So what does a “40% chance of rain tomorrow” really mean? Does it mean 40% of the area will be rained upon? All forecasts are organized by areas, so that seems a reasonable take. Perhaps it means it will rain 40% of the time. So in a 24-hour day you’d get 9.6 hours of rain. I don’t like that one, and I’m not sure why, but I could see someone interpreting it that way.

I would always follow up my initial foray into probabilities with another version of “40% chance of rain tomorrow.” I told my students that if we could replay the day 10,000 times it would have rained in 4000 of them.

That’s a little whimsical, we get to play god and mess with time, but computer simulations allow us to think like that. You have to check your models against nature, so you better go back and see how well they did!

Turns out the **National Weather Service** has an official definition for this so we don’t have to fret (**PoP** is **P**robability **o**f **P**recipitation):

Mathematically, PoP is defined as follows:

PoP = C x A where “C” = the confidence that precipitation will occurin the forecast area, and where “A” = the percent of the area that will receive measureable precipitation,somewhereif it occurs at all.So… in the case of the forecast above, if the forecaster knows precipitation is sure to occur ( confidence is 100% ), he/she is expressing how much of the area will receive measurable rain. ( PoP = “C” x “A” or “1” times “.4” which equals .4 or 40%.)

But, most of the time, the forecaster is expressing a combination of degree of confidenceandareal coverage. If the forecaster is only 50% sure that precipitation will occur, and expects that,if it does occur,it will produce measurable rain over about 80 percent of the area, the PoP (chance of rain) is 40%. ( PoP = .5 x .8 which equals .4 or 40%. )In either event, the correct way to interpret the forecast is: there is a 40 percent chance that rain will occur at any given point in the area.

I suspect a lot of folks will find that unsatisfying, but this mathematical view is at least a precise definition. And it seems to cover both the “area” part and the “time” part.

So what does it all really mean? I think a bunch of meteorologists get together at lunch and look out the window and argue about whether or not it will rain tomorrow. Finally they agree to state it as a probability, and experience tells them how often they’ve been wrong. So “40% chance of rain tomorrow” really means “we get this right four out of ten times!”