Gur Huberman points us to this page [link fixed] from the National Weather Service, “Explaining ‘Probability of Precipitation,'” which says:

Probability of Precipitation = C x A where “C” = the confidence that precipitation will occur somewhere in the forecast area, and where “A” = the percent of the area that will receive measureable precipitation, if it occurs at all. . . . 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.

So now you know.

**P.S.** Commenter Adede links to an amusing xkcd cartoon on the topic. The cartoon makes it look very mysterious and even unknowable. But the concept of “chance of rain” really can be defined precisely; see above!

**P.P.S.** Separate from the question of what these probabilities are intended to mean, is the question of how people interpret them. To this end, Eric in comments points to this 2005 article, “‘A 30% Chance of Rain Tomorrow’: How Does the Public Understand Probabilistic Weather Forecasts?”, by Gerd Gigerenzer, Ralph Hertwig, Eva van den Broek, Barbara Fasolo, and Konstantinos Katsikopoulos. Interestingly enough, Gigerenzer et al. consider several possible interpretations of the statement “30% chance of rain,” *none of which* correspond to the interpretation given by the National Weather Service above.

Obligatory xkcd:

https://xkcd.com/1985/

Also, the link in your post points to the admin login.

Correct link:

https://www.weather.gov/ffc/pop

Link fixed; thanks.

I honestly thought it meant that the estimate comes out of a model calibrated so that out of all days in which the forecast was 30% rain, it actually rained 30% of the time

That would be the frequentist way to interpret it. A Bayesian interpretation of the probability of a singular event like “rain on 13 Dec 2019 at location x” would be the following:

We have partial information about the weather, but there’s lots of things (measurements? parameters?) we don’t know. 30% of those unknowns lead to rain, while 70% don’t.

Note that if weather forecasters run lots of weather model simulations and predict rain when 30% of the simulations give rain, then in effect, they’re using the later bayesian version than the former frequentist one.

Incidentally, trying to be wrong 30% of the time is a bizarre goal for prediction.

I looked at the link. It interprets a 40% chance of rain, not 30%.

There is a little literature here on how people generally think about probabilistic forecasts. As in here:

Gigerenzer, G., Hertwig, R., Van Den Broek, E., Fasolo, B. and Katsikopoulos, K.V., 2005. “A 30% chance of rain tomorrow”: How does the public understand probabilistic weather forecasts?. Risk Analysis: An International Journal, 25(3), pp.623-629.

> there is a 40 percent chance that rain will occur at any given point in the area

That may be correct in the sense that if we sample randomly a point in the area, there is a 80% chance that the point happens to be in the area were it rains if it does rain (a 50% event). But unless we assume that the rain is distributed uniformly we cannot say that the probability of rain in every point in the area is the same.

It’s similar to the interpretation of many frequentist methods which is valid “pre-data” but not “post-data”. The probability as described may be valid for any unspecified point, but not for any _given_ point.

Even more problematic is the assumption that consumers of this information (people) are uniformly distributed in the area. I mean, if it’s going to rain on top of a mountain which takes up 30% of the area, but all the people live at the base of the mountain taking up 10% of the area… they are almost 100% chance NOT going to get rain, even if there’s a “30% chance of rain” meaning “it’s 100% chance going to rain on top of this mountain”

I agree completly. “There is 40% probability that an unspecified recipient of this forecast will experience rain” and “there is 40% probability that a given recipient of this forecast will experience rain” are different things and arguably more relevant than the “geographical” forecasts.

I actually think they should stop giving percentage chance of rain, and start giving expected rainfall and a 95% hpd interval “We expect .6 inches of rain with 95% chance of between 0 and 1.3 inches” is much more useful than “30% chance of rain”

NOAA forecasts for Quantitative Precipitation and Snowfall as an experimental Probabilistic Forecast feature sound similar to what you’re describing.

I like that idea!

PoP refers to measurable precipitation, which is >= 0.01 inch in the chosen time slot (https://www.weather.gov/bgm/forecast_terms). That may not be the immediate interpretation when someone reads a forecast of 0.5 inches of rain in the next hour with 10% probability.

To add to this – the hourly forecast is (at least for the BBC in the UK) backward looking, as discussed on [More or Less](https://www.bbc.co.uk/sounds/play/b0b3fz4c) (first item). The example they give is that the forecast at 2pm refers not to the hour 2-3pm, but to the hour 1-2pm.

The explanation in itself has a nice analytical underpinning in that it is based on the fact that to verify your model you would look backwards.

My forecast is that there is a 99% chance that exactly how the public understands a forecast of “25% chance of rain” won’t ever matter, because whatever misconceptions they have are small enough to be totally irrelevant.

Even if it was a tsunami forecast it wouldn’t matter much, since if there’s a “10% chance of a 90ft tsunami” in the next four hours, most people would be happy to move inland for that period.

What’s really interesting to me is the idea that it’s now the responsibility of the weather service to somehow ensure that it delivers the precise wording that compels people to do what they’re “supposed to” do for any given weather event.

Interesting, but this doesn’t answer the question in the first panel from the xkcd comic.

If there is a forecast for each of the next 5 hours, and there is a probability of precipitation of 20% in each of them, what does it mean for the whole 5 hours? Naively just adding up the probabilities gives 100%. Assuming independence, we get 67%. But independence seems unrealistic. So is the PoP for the whole 5 hours close to 20% itself?

I would contest your claim that “[xkcd] makes it look very mysterious and even unknowable.” The XKCD raises very real ambiguities in how one should interpret a set of predictions given according to the precise NWS definitions.

For example, if I say “there is a 50% chance that I will quit my job, and a 50% chance that I quit smoking”, I have made two predictions that can be precisely defined, but I haven’t specified if these events are independent, correlated, or completely codetermined. I haven’t specified a proper joint probability distribution.

That is exactly the ambiguity that the XKCD is pointing out: each prediction is precise, but the set of them together is ambiguous.