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#1




What are the odds of rain?
So the weather app tells me the chance of rain is:
1PM: 50% 2PM: 50% 3PM: 50% What are the odds it will rain somewhere in the 1:00, 2:00 or 3:00 hours? The outcome of rain is an independent event, so one could calculate the chance of rain within any of these 3 hours is 87.5%. Or should I read the weather forecast to mean there is a 50% chance throughout that entire three hour timeframe? 
#2




Rain is not an independent event, and I'm not sure why you would think it is. There is not enough information to calculate the total probability from that information.

#3




The latter. These forecasts are for joe public, not professors of mathematics.
50% means that it is a coin toss as to whether it will rain or not. Last edited by bob++; 07192016 at 12:19 PM. 
#4




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What more info do you want from the weather app? 


#5




I think if it rains at 1pm, that is likely to make it either more likely or less likely that it will then rain at 3pm, but I don't know which. Nor, apparently, do professional meteorologists, much of the time. In other words, I think there are too many factors to say.
I personally don't pay much attention to the probabilities. Where I live (SW England), they are rarely accurate in my experience. The other day it said 0% chance of rain at 7pm, but there was some (with me looking at that forecast only a few hours before). So they're basically useless for any kind of forward planning. 
#6




As The Master explains it "What does it mean when 'X percent chance of rain' is predicted"?:
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Last edited by watchwolf49; 07192016 at 01:29 PM. 
#7




Maybe it means that there's some big storm coming, and it might miss us, but if it doesn't, it'll rain for three hours. In that case, if it's still dry at 1:00, then it probably will be all day, because it meant the storm missed. On the other hand, maybe they're sure the storm is coming, but they're not sure precisely when it'll hit. In that case, if it hasn't started yet by 1:00, it's even more likely that it'll be raining later. Either case is plausible, as is any combination of the two.

#8




The hourly estimates aren't to be taken as actual probabilities. They start with the probability for the day, actually half day in most circumstances. If all times of day are equally likely, then every hour is assigned the same probability. But often certain hours are more likely to get rain due to predicted changes in weather conditions so the odds are moved up or down accordingly. So if it's like to be dry in the morning, the odds are low while if it's going to be rainy in the after noon, the odds go up quite a bit (with most of the percentages being alloted there).
And then they fudge the numbers some more since they still don't make a lot of sense. It can get pretty far from real statistics. I just checked this with my local weather.com forecast comparing today's chance with the upcoming hourly chances. Yeah, it doesn't make any sense at all unless you see how they're fudging things. 
#9




Weather prediction is not a science. It is a Black Art.



#10




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I'm inclined to say that if asked what are the chances of rain between 1PM  3PM, the best answer seems to be 50%. 
#11




Predicting rain is chancy stuff ... for rain to form in 5ºC air, the relative humidity must be 100% ... but if that same air is 5.5ºC, then RH will be 95% (abouts) and it won't rain. Hell, clouds won't even form so if the forecast is a 90% chance of rain, there's a 10% chance it'll be sunny and warm ... and it's all about a half a degree higher or lower ... tricky stuff if you don't sell your soul to the Weather Demon.

#12




Calcualte the negative, to get the simple probability of all being met The chance that it will rain in one of those periods equals one minus the chance it will not rain in any of them. So, the chance of no rain is 1pm0.5, 2pm0.5 and 3pmo.5. Multiply those and get 0.125 as the chance of an absence of rain in all three, or 0.875 as the chance at least one of them will have rain.
As a better illustration, say the chance is 20% this evening, 50% overnight and 30% tomorrow morning. The chance there will be rain at least once by tomorrow noon is 1(0.8x0.5x0.7) = 10.28 = 0.72. or 72%. We've calculated the chance it will NOT rain (80%, 50% and 70%) is 28%, so the chance it WILL rain is 72%. Last edited by jtur88; 07202016 at 12:26 AM. 
#13




Taking a look at weather radar and weather radar predictions tells usually a lot more about rain than just looking at the hourly rain possibilities. If you see a bunch of high intensity small rain areas wandering around there's a chance of either no rain or a lot of it, but if you see a wide rain front approaching then it is more of a question when it arrives than whether it will arrive at all.

#14




jtur88, that assumes that all the probabilities are independent, which we've already pointed out probably isn't true.



#15




You can't take each of those predictions as an independent event. To take it to an extreme, suppose the forecaster gave predictions every minute during those three hours instead of every hour. Almost certainly all those predictions would be identical (if you think otherwise, what do you think they would be?).
1:00 50% 1:01 50% 1:02 50% ... 2:59 50% 3:00 50% Using jtur88's reasoning, there are 181 independent events with P=0.5 for each, so the probability of no rain is (0.5)^181, or 3 x 10^55. So rain is essentially guaranteed, simply because more predictions were made, which is an absurd conclusion. Mark Last edited by markn+; 07202016 at 11:33 AM. 
#16




A lot of times when they say 50% chance of rain accumulation is only .001 inches. That's not very much rain.

#17




Strictly speaking, these rain probabilities are only generated in 6 hour time intervals. We can of course interpolate these probabilities every hour or every minute, but the output of the numerical models are every six hours.
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#18




If I pick two days at random, one of which had rain at 1:00 and the other of which didn't, and asked you to bet on which day had rain at 3:00, which day would you bet on?

#19




I've often wondered about this, and we don't yet seem to have a clear answer to the OP. As the OP and others have said, the stated probabilities for rain during different lengths of time interval in popularized weather forecasts are quite obviously mutually inconsistent under any assumptions of dependence between consecutive periods.
watchwolf49 gives a cite that forecasts are calculated in 6hour intervals, but since meteoroligists can actually do math, it does not necessarily follow that the probabilities that they distribute from their forecasts apply to 6hour periods. I'd really like to know a few things: (1) What exactly constitutes "rain" in a given period in the models? (If the threshold for a "rain" event is higher for a longer period, this might go some way to explaining the apparently inconsistent probabilties.) (2) What's the best primary source of weather forecast data, where the probability is actually correct  i.e. where if the probability for a given period is stated as 50%, there really is an expectation under the model that rain will occur half the time in the stated period? (3) What kind of screwy calculation is done (and by whom) to produce the nonsensical hourly probabilities shown in popular forecasts? Last edited by Riemann; 07202016 at 02:56 PM. 


#20




Given the lack of independence, the best you can infer is that the probability is somewhere between 50% and 87.5%. Either that or you need a source of information "upstream" of the weather app  you need access to the information that led to the information in the app.

#21




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(2) Some of the best and most accurate weather forecasts come from voodoo witch doctors ... [giggle] ... yes, that's being sarcastic ... but seriously, let's not confuse "best source" with any source that's in anyway "good". We do the best we can and that's the best we'll get. For example, we take our current weather conditions and search our historical data and find the same conditions 138 times. Of those 138 times, it rained within 6 hours on 16 occasions. We would then forecast a 10% chance of rain in the six hours following our current time. Yes, 10% here actually means anywhere from 5% to 14.9%, socalled accurate to one significant digit. So, as long as we're clear that "best" source is "not very good either", then really any are fine. One BIG difference comes in whether the source is commercial or noncommercial. The noncommercial sources tend to be more honest, more willing to say they don't know and so they're just guessing. So the National Weather Service or World Meteorological Service. When I lived in Iowa, we had one local station's weather guy always forecast exactly what the farmers wanted. If it was running dry, he'd forecast rain; if things were too wet, he'd forecast sunny ... it was awful. But you know, the guy had products to sell and he sold more if he said things would be great in a couple days, "brought to you by The Johnson City Seed Company, where all news are good news". (3) They interpolate (see Wikipedia article). We find the inbetween result by weightbased averaging. So if 2 x 2 = 4 and 3 x 3 = 9, then 2.5 x 2.5 ≈ 6.5. Similarly, if rain is 40% probable in the first six hours, 60% probable in the next 6 hours, we can approximate that it will be 50% someplace inbetween. ≠≠++≠≠ Yes, meteorologists can do the math, here's the formula caste as partial differential equations, NSFH twoclick rule applies (DO NOT let your teenage kid see you looking at this or you WILL be doing a shitload of algebra homework next school year) SPOILER:

#22




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By (2), I meanst a source for the unadulterated forecasts that the models are producing  i.e. what comes out of the NOAA or other models, before being screwed up and misrepresented for popular presentation. The NWS website is the most credible source that I'm aware of, but it doesn't show very much detail of a probabilistic nature that I can find. Re (3)  so, ok, then so long as we know the time period for which the probability is actually a probability (rather than just an mathematically invalid manipulation), then at least we know where we stand. Can you direct me to a primary source for NOAA forecasts that shows 6hour forecast blocks with associated probabilities, if that's the way the real probabilities are generated in the models? many thanks Last edited by Riemann; 07202016 at 05:05 PM. 
#23




I have this link from the NHC "NHC Forecast Model Information". This page also gives links to regular forecast models. Be warned:
"The National Hurricane Center does not generate a graphic of the guidance models it uses to produce its forecasts. Such graphics have the potential to confuse users and to undermine the effectiveness of NHC official tropical cyclone forecasts and warnings." Keep in mind that some of these models are copyrighted and leased to the NWS under the conditions the NWS doesn't publish the detailed results ... however I seem to remember a NOAA site that actual allowed the user to download some of the public domain models. I imagine you'll need a stack of MacMinis to get the flops you'll need to get a six hour forecast within six hours. I don't know what you mean by "a primary source for NOAA forecasts that shows 6hour forecast blocks with associated probabilities" if none of the links I've provided already serve that purpose. 
#24




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So I'd be interested to know some place where I can see the true mathematically coherent forecast probability distribution that's generated by the NOAA model for a given location. Perhaps that's somewhere on the NOAA site that you linked, but I haven't been able to find it so far. Where can I see a valid probability distribution for rain in some major city over some time period in the near future? Pick any place and time period that you like, I'd just like to see what a real forecast looks like. Then maybe we can map that to what the hell the news services are doing with that to produce what they feed to the public. 


#25




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A commercial news source will flat lie to you if it sells more Ford F150 4x4's with the snow plough attachment on the bumper. How do they come up with these probabilities ... they lie ... simple. ≥≤≥≤≥≤ Again, you seem to not understand probability ... We have 138 trials, and 16 are positive ... why don't you understand this is a 10% probability (accurate to one significant digit)? Last edited by watchwolf49; 07202016 at 06:43 PM. 
#26




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It's a very simple question  do the NOAA forecasts in original form give a rain probability (something that really is a probability) for the binary event of >0.01" rain within a 6hour window? Quote:
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The issue under discussion is the time periods for which probabilities of binary events are forecast, and the obviously invalid methods that news services are using to convert these to probabilities for shorter or longer time periods. Last edited by Riemann; 07202016 at 07:33 PM. 
#27




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The statistics here are simplistic, the big computers are needed to search huge databases. If they get 528 hits and it rained 374 times ... that's 70% probability. 
#28




Please, I understand The Master's article in its entirety. Although I am old and grumpy these days, I do have a degree in math and a lapsed commercial pilot's license. I'm not seeking to be educated in the principles of meteorological modeling and forecasting or how to divide one number into another and express the answer to one significant figure.
I laid out my questions quite clearly in my prior comment. Since you seem to be connected with NOAA, I thought you might know the answer. Do you understand the problem with obtaining the probability of a binary event for differing time periods, as laid out by the OP and others such as markn+ above? I'm asking, quite simply: What form do the probabilistic elements of NOAA forecasts take before they are manipulated by the news services? What is the time period for which the probability of rain is given in a standard NOAA forecast? I gather that I would need to pay a lot of money to see an actual NOAA forecast, but perhaps you have an example of an old one to see the form of an NOAA forecast? I already know how to see the NWS forecasts and aviation forecasts. Then, what are NWS and the news services doing to obtain probabilities for time periods of different lengths? The NWS forecasts just say a probability for "today" and "tonight", what exactly does that mean  a 12hour window? Are the NWS probabilities at least derived by a valid methodology from the NOAA forecasts? If you don't know the answer, that's fine, but please just say so. For Jeebus sake, stop linking to things like the Wiki article on interpolation. Last edited by Riemann; 07202016 at 09:38 PM. 
#29




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What do the news services do with the data to get the one hour probabilities? I don't know. It's certainly not math, and I serious doubt it's science ... and I'm sure every news services does its own way. However it's made up, it's not coming from the NWS and should be regarded with caution. 


#30




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If the first thre batters in an inning are batting .330, .240 and .220, what is the chance they will go three up three down? 1  (.67 x .76 x .78), or 1  .397, or .603. Last edited by jtur88; 07202016 at 11:34 PM. 
#31




Back in Grad School, my very cool statics professor ( Professor Chen ) explained that statistically over time, the weather changes every fourth day.
The way he explained it, if it is sunny for 3 straight days, statistically the weather will be something other than sunny on the fourth day. Yes, there were a lot of complex calculations behind forecasting the weather... but... if you followed that very simple rule of thumb you would be right most of the time. As a rule, I found it so simple that I never forgot it... and the results of its application so close to always acurate that it was almost laughable. I miss Professor Chen. Great Professor. Great Guy.... 
#32




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Last edited by Riemann; 07202016 at 11:52 PM. 
#33




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And we have no clue what the hourly "probabilities" given in popular forecasts are even supposed to mean, except that in a qualitative sense they do indicate at what time of day any rain is more likely to occur. 
#34




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Mark 
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