The actual chance of rain during the day
so often, we see an hour by hour prediction where the chance of rain, might be, for example, 20% at 8 am, 40% at 9, 40 again at 10, etc.... to 10% at 4 pm.
I am thinking that the chance of rain sometime during the day is far higher than the chance at any particular hour, but what math would be appropriate to estimate the overall probability? One way might be to multiply the chance of NO rain for each hour throughout the day, and subtract the answer from 100. Is there a better way? 
That would be the way to do it if the probability of rain in each hour were independent of the probability in every other hour. But in real life that's definitely untrue. For example, suppose there were a 25% chance of a storm hitting your area in one day, and if it did hit, it would rain all day long. Then the reported probability of rain might be about 25% in each hour, but that certainly doesn't mean that the chance of rain any time in the day is 1.75^{24} = 99.9%.

Frustrating prior thread here:
http://boards.straightdope.com/sdmb/...d.php?t=798848 So far as I can discern, the NOAA/NWS provide forecasts for certain time windows that are coherent probabilities, see the last post there. Then the popular news services use invalid methods (I don't know exactly what) to obtain their "probabilities" for shorter or longer time periods. I think that the numbers that you see on popular news services for hourly probabilities are essentially nonsense. 
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My local TV's weather App gives a daily forecast and chance of rain.
For example, next Wed 4/26 there is a 100% chance of rain with Afternoon thunderstorms. Meaning, rain will fall in my local area sometime during that day. Most likely Wed afternoon. Tuesday, I'll be able to click Hourly (for Wed) and get percents throughout Wed. I've learned (the hard way) if it says it'll rain at 2pm (60%) then I need to finish my errands by 1 pm or get drenched. It's absolutely astounding to know the hour when rain is expected. I can go to the radar map and see the storm 60 miles to the west of me. I find this incredible. We used to listen to the Farm report and weather at 5:30AM before heading down to the barns to work. That one forecast is all we had to plan our day. We might be bush hogging the pasture and get hit by a thunderstorm. My Uncle was really good at looking at the sky and spotting storms. We'd get our equipment in the barn just before they hit. 
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The standard Prob. of Precip. that the NWS uses is confidence x coverage. So if the NWS says that the PoP is 100% on Sat. afternoon for a given area, they are 100% confident that 100% of the forecast area will see rain at some point during the period in question. If the hourly probability of precip that TWC offers says something like 1pm: 40%, 2 pm: 50%, 3 pm: 90%, 4 pm: 90%, 5 pm: 75%, 6pm: 30%, that should mean (assuming they agree with NWS that the coverage will be 100%) that they are not certain exactly when in the time frame involved the rain will occur. Confidence level goes up as the front's expected arrival time gets closer to what most of the models show, and drops off as time passes beyond that. Do you think that this is not true? 
You have given a set of perfectly plausible numbers, if they were doing an actual calculation of probabilities.
But look at some actual numbers. Here is the forecast right now for Atlanta, GA. Convectional showery weather. The forecast for rain at any time on Monday is 50%. For the same period, the hourly "probabilities" are: 7am 40% 8am 45% 9am 50% 10am 50% 11am 45% 12pm 45% 1pm 50% 2pm 45% 3pm 45% 4pm 40% 5pm 35% 6pm 35% 7pm 40% How can that possibly make sense? That could only be true if the correlation between each hourly period were essentially 1. 
They are predicting when a storm front will be over your area. Looks like you'll get rain most of that day until 5pm. It'll decrease into the evening.
That's a difficult example. I typically see 9 AM 20% 10 AM 20% 11 AM 25% 12 PM 30% 1 PM 45% 2 PM 55% 3 PM 25% 4 PM 20% Clearly indicating they expect the storm front to pass through between 1pm to 3PM. It might roll in a little early around 12:30 PM or linger after 3 PM. 
Our favorite local weatherman here often reminds us that, even though it will likely rain sometime during the day, the chances are, it will be not raining most of the time, and that outdoor events might be interrupted, but not washed out. In our area, we see a lot of rain cells coming up from the southwest, headed toward the East Coast. Some days we get hit by several of them, other days they miss us completely. The story, though, is essentially the same, the same 30 or 40% chance of rain every hour from mid morning until evening. Nighttime predictions seem a bit more likely to be accurate.
Anyhow, surely they can, based on history if nothing else, suggest that the actual chance of rain at any given hour is less than half the chance of rain sometime during the day, couldn't they? 
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For the Atlanta area, there is 100% confidence that roughly 50% of the area will see showers. Broken down into hourly segments, the confidence remains close to 100%, and the coverage is around 50% at any given time. In other words, scattered showers are predicted with almost 100% certainty. 
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I still lean toward the view that they are not really calculating probabilities. What I'd really like to see is some kind of clear explanation of what they are doing and what the numbers mean, but as you can see from that other thread, nobody seems to know, and nothing is documented on their website (or any website that shows hourly numbers). 
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Our local weather is notoriously unpredictable, and the NWS will flat out state that the models are everyplace and there's no consensus ... but average weather for a given date is such and that's the forecast for no other reason other than that's as good a forecast as can be had ... you'll never hear that from a commercial weather report, or it would present Jiffy Burger's in a bad light ... As long as these outlets don't say how they calculate these hourly rain probabilities ... then it's fair to assume it's just smoke and mirrors ... or that they're just making it up ... There's a few people who report the weather who have a good idea of the principles involved ... but most are just reading NWS copy dumbed down for the masses ... 
This statement of probability is not like rolling dice, where the probability of getting a certain number is known with statistical certainty that a sufficient number of rolls would converge to. It is, rather, based upon the observed frequency of rain occurring with conditions similar to those predicted by the weather model, e.g. at a certain temperature, relative humidity, and warm front coming in, rain is observed to occur xx% of the time. There is some additional conditional probability from cumulative predictions of precipitation such that a day in which each hour is predicted to have a 50% probability of rain may have a cumulative probability of 60% or 70%, but it is all based upon largely empirical models and some simplified atmospheric behavior modeling.
Stranger 
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The issue is that when you look at the numbers that the likes of weather.com produce, their hourly probabilities often look much too high to be consistent with the daily probabilities, i.e. they imply an implausibly high hourtohour correlation. I'm open to being proven wrong, but until weather.com et al explain what they are doing and precisely what their hourly "probabilities" are supposed to mean (it's not documented anywhere that I can find), I remain skeptical. 
I have no idea what Weather.com does or what they base their predictions on. Weather Underground, on the other hand, tends to line up reasonable well to NOAA predictions.
Stranger 
Here's the updated forecast for Atlanta Georgia:
Rain at any time on Monday = 40%. Simultaneous hourly percentages: 7am 30% 8am 40% 9am 40% 10am 45% 11am 40% 12pm 45% 1pm 45% 2pm 25% 3pm 35% 4pm 35% 5pm 25% 6pm 20% 7pm 25% 
Remember that even when we're talking about confidence x coverage, we're still talking about the probability of a binary outcome for each "patch" within the area  i.e. over what proportion of the area is it expected to rain at all in the given time period (see below).
Given that, how can the numbers above from weather.com possibly make sense? How can ther be three individual hours when confidence x coverage is greater than the figure for the entire day? http://www.weather.gov/ffc/pop Quote:

Best I can interpret is that the allday number is the probability that it'll be raining at any randomlyselected point during the day. Thus, if it were known with 100% certainty that 100% of the area would be dry all morning, and rain from the stroke of noon to the end of the day, then the daily number would be 50%.
But then, that interpretation is inconsistent with the numbers in post #8. 
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In the case Chronos presents is somewhat fanciful ... we've taken the 6 am weather data, plugged into the computer model and let 6 hours pass in the model and our result is rain starts right at noon ... upon consulting the historical record we'd have to find that under the 6 am conditions, it has never rained earlier than expected in the past 100 years of record keeping ... because if in 10% of the historical cases it did rain earlier than expected ... then the forecast would say "10% chance of rain before noon, then 100% chance of rain after" ... things really do change that quickly ... One thing the NWS does that no one else seems to bother with is they provide a detail discussion of the forecast, where the forecaster will state which of the models give what results and which model run the forecaster is relying on ... "The 12z GFS sped up precipitation timing closer to the 00z EC solution and suspect most of the region will be wet by at least late Tuesday afternoon...if not sooner." We have to have a crew of professional forecasters to make these predictions by the hour ... and racks of powerful computers to get the results ... I honestly don't think The Weather Channel has invested that much money ... 
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