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Another issue that affects how accurately weather models are seen is a difference in what they output and what is reported. The output of the models is a series of 2D maps of various variables and how they change over time and space. The models will predict that conditions will form for cloud formation and will lead to precipitation at a certain point. They are pretty good at predicting that part. Where it starts to get less accurate is determining where those things will happen and when with specificity. They'll be pretty sure that there will be rain from this particular system, but it might move north of city x, go right over it, or go south of it.
So that 40% chance of rain is actually "99% chance it rains, but 39.5% chance it rains here and 59.5% chance it doesn't rain here but somewhere else nearby instead".
My appreciation for what they do increased after I started using windy.com, which gives the map of predictions over time instead of "here's what will happen in this city".
Oh and weather patterns can be smaller than cities, too. That 40% chance of rain could even mean "40% of the city will be rained on". On cloudy days, you can often look around and see rain in the distance in various directions around you, sometimes it passes over you sometimes it doesn't.
Right, that's exactly what it means, by my understanding. A weatherman can't predict the chances that a particular individual will experience precipitation, but that's what the average person immediately thinks when they look at a weather forecast.
What a weatherman can do though, is predict how much of a particular area may experience precipitation, based on measurable things like cloud pattern shape and size, wind speed and direction, geography, etc. Once you realize that, it's actually kind of intuitive that precipitation chances are reported as "percent of an area that will experience precipitation", and not "percent chance that I will experience precipitation".