The forecast says "40% chance of rain." People read that in many ways, most of them wrong. The National Weather Service defines it precisely, and the definition connects to the ensemble forecasts you just met and to probability from math.
Key idea
The probability of precipitation (PoP) is the chance that measurable precipitation (at least 0.01 inch, about 0.25 mm) falls at any given point in the forecast area during the forecast period. The National Weather Service computes it as $$\text{PoP} = C \times A$$ where $C$ is the forecaster's confidence that rain will occur somewhere in the area and $A$ is the fraction of the area expected to get it.
Why it works
The forecast has to answer the question you care about: "will it rain where I am?" If a forecaster is sure it will rain ($C = 100\%$) but only on 40% of the area, your spot has a 40% chance. If she is only 50% sure rain will develop, but if it does it will cover 80% of the area, your chance is $0.5 \times 0.8 = 40\%$ again. Both forecasts say 40%, and both mean the same thing for you: in situations like this, rain falls at a given spot about 4 times in 10.
Worked example: Computing a PoP
A forecaster is 80% confident that showers will form, and expects them to cover half the county if they do. What PoP goes in the forecast?
- $C = 0.8$ and $A = 0.5$.
- PoP = $0.8 \times 0.5 = 0.4$, a 40% chance of rain.
Answer: 40%
Worked example: Checking the forecasts
Over a year, a forecaster gave a 30% chance of rain on 60 days. Rain fell on 19 of those days. Were the forecasts well calibrated?
- A 30% forecast should see rain on about 30% of such days: $0.3 \times 60 = 18$ days.
- Rain fell on 19 days, very close to 18.
- Yes: the 30% forecasts meant what they said. A day with 30% that turned rainy was not a "wrong" forecast.
Answer: Yes, well calibrated
Common mistake
Thinking a 40% chance means rain for 40% of the day, or over 40% of the area. It means a 4 in 10 chance that measurable rain falls at any given spot in the area during the period. Check: 40% of the area with full confidence and 80% of the area with half confidence give the same 40%.
The chance of staying dry is $1 - \text{PoP}$: a 40% chance of rain is a 60% chance of no measurable rain at your spot.