weather.com gives both daily and hourly probabilities of precipitation. For instance, tomorrow here has a p.o.p. of 20%; but each hour during that day has a p.o.p. ranging from 10 to 40%.
If each hour’s p.o.p. is independent, then the overall probability for the day should be 98%, not 20%, just as the probability of rolling a 6 with a normal die is 16.7% and the probability of rolling a 6 at least once in 24 rolls is 98.7%.
So, what gives? How do the folks at weather.com come up with these numbers?
It doesn’t work like that. Think of it as a single event (a rainshower). It has an overall probability of happening during the course of a day, but it also has a probability distribution–basically a Gaussian graph–wherein it has less chance of occuring at some times of day than others. The important point is it’s only going to happen at one point during the day, so the probabilities don’t add up like you suggest.
Take a coin flip. 50% chance it’ll be heads on the first flip, second flip, third flip and so on. If the chances were cumulative I’d HAVE to throw heads on the second flip for certain but we all know this isn’t the case.
Another way of looking at your example is that there is a 20% chance of rain. That means there is an 80% chance of no rain. The folloing hour there is a 40% chance of rain thus a 60% chance of no rain. We already have a 140% chance of no rain.
As to the chances of rolling a 6 at least once in 24 rolls being 98.7% say 6 represents rain. Say 1 means no rain. You also have a 98.7% chance of rolling a 1 at least once in 24 rolls so you have a 98.7% chance of no rain.
Whack-a-Mole, the OP is not talking about “cumulative” probabilities. I think maybe you misunderstood or something.
Here’s an example of a weather forecast:
Chance of rain in the morning: 50%
Chance of rain in the evening: 50%
Chance of rain overall in the day: 50%
But if the two events were indeed independent, the final probability should be 75% (not 100%, like an unrealistic “cumulative” probability would suggest).
I think what’s going on is the forecasters take the average probability per unit time over the course of an hour or whatever, and multiply it by 1 day. That way you can have a so-called 48% chance of rain between 15:00 and 16:00, while still having an actual 2% chance of rain in that interval.
The OP understands all this, of course, and is asking for confirmation or denial if that’s the way it’s actually done. I myself can only speculate.
Perhaps it’s something like this? Maybe their daily % chance of rain is the rounded average of all the hourly chances of rain?
I once had a statistics professor who worked for NOAA at one time doing weather statistics and prediction. I think he said all we can really tell for sure is that if it rained today, it will probably rain tomorrow.
Here in Florida it rains when the sun is shining, I wonder how they categorized that phenomenon? It reminds me of the old Creedence song … “have you ever seen the rain coming down on a sunny day?”. Well, yes I have
Yes, that’s what I’m driving at … but since they do use numbers, there must be some probabilistic calculations underlying them somewhere (unless they’re just rolling dice).
And it is not at all clear to me that, say, the p.o.p. at the 1 am hour is not independent of the p.o.p. at 3 pm. There could easily be separate weather system, after all.
I suppose I could try the empirical approach and note those hours during which it does rain (and I’m awake to notice it) and compare my findings with weather.com probabilities. (Or perhaps my kids need science projects for this coming year.)
I do believe the models they use to predict the weather are the most complex mathematical models know to man. They use the biggest, fastest computers in the world to make their calculations. Many scientists and mathematicians have contributed to this model over the years. I’m sure if you or I were to see this model face to face we could only say … huh?