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?
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.
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.
The probability of rain in the 1PM hour doesn’t effect the probability of rain in the 3PM hour, does it? Or does it? They’re certainly not mutually exclusive, meaning it could rain in each of the 3 hours.
What more info do you want from the weather app?
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.
As The Master explains it “What does it mean when ‘X percent chance of rain’ is predicted”?:
So in to OP … at 1 pm, half the time it rained, half the time it didn’t when the weather was the same. At 2 pm the same thing; half it did, half it didn’t; etc etc etc … ad nausium
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.
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.
Weather prediction is not a science. It is a Black Art.
Interesting.
I’m inclined to say that if asked what are the chances of rain between 1PM - 3PM, the best answer seems to be 50%.
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.
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 1pm-0.5, 2pm-0.5 and 3pm-o.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) = 1-0.28 = 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%.
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.
jtur88, that assumes that all the probabilities are independent, which we’ve already pointed out probably isn’t true.
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
A lot of times when they say 50% chance of rain accumulation is only .001 inches. That’s not very much rain.
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.
“Weather Models” from NOAA.
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?
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 6-hour intervals, but since meteoroligists can actually do math, it does not necessarily follow that the probabilities that they distribute from their forecasts apply to 6-hour 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?
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.