It's specific to the Pacific Northwest (say, a few hundred miles around Seattle), but I've learned a whole lot about the general topic of forecasting by following https://cliffmass.blogspot.com/ . I don't know how much access you'd have in other areas to the underlying measurements and modeling that go into the summary weather reports you can easily get in apps or on media sources.
I don't know of anywhere you could get such information, short of analyzing the ensemble of simulation outputs yourself. With typical ensemble sizes of "a few" to "a few dozen", you probably can get useful confidence intervals but probably can't get useful conditional probabilities.
(specifically: you'd want to search for your national weather service's "Thredds data service", then get the "opendap" link, and use xarray.open_dataset() on that URL... with lazy loading, the data are usually TB+)
The traditional weather forecast consists of summary statistics over saved timesteps of detailed simulations, which run forward from a best-possible reconstruction of the current state of the atmosphere. Data-assimilation, or "hindcasting"/"nowcasting", is itself a neat trick, and the dual of forecasting - you have past and present observations; you have a model of the system dynamics; you can sample from plausible system states which are compatible with observations or even solve for the most-likely state given observations (including subsequent observations). I don't think enough people realize that we can be so much more confident about the details of the weather last week than today, even in remote places where nobody was watching!
For a variety of reasons I've been looking at weather forecasts lately, a lot more than I ever used to, and I'm more and more aware of both how useful and how incomplete the ranges of temperatures and probabilities of precipitation are. Like, if there's a chance of rain rising from 0% at 3pm towards 100% at 7pm, that's pretty clearly "it's going to rain, we're just not sure when it'll start." But if there's a 30% chance of rain on each of 5 consecutive days starting 4 days from now, is that "It'll probably rain, but we don't know when or for how long" or "a storm may or may not form at all, and if it does it may or may not head your way"? And what are the error bars on high and low and expected temperatures, in the event that I care a lot about whether we cross a particular temperature threshold (for example, above or below freezing)? Similar questions on error bars for wind speed and direction forecasts, both sustained and gusts.
I don't really see any good way to convey this in typical forecasts in a way most people would understand or care about, but is there anywhere someone who isn't at the level of a meteorologist can go for this kind of info or something like it?