Dear 100 Hour Board,
My question is about meteorology and probability. From my limited understanding of meteorology, I understand that the probability of precipitation is the product of C and A, where C is the confidence that there will be precipitation somewhere in the forecast area and A is the portion of the area that will receive appreciable precipitation. Ultimately, though, what this means for the consumer of the forecast is that in a given area, he or she has an X% chance of getting precipitation.
My question arises in the difference between hourly and daily forecasts. To make things simple, if an hourly forecast indicates, for example, that on a given day there are six consecutive hours each with a 30% probability of precipitation, then (now coming from a mathematical standpoint) the probability that there will be no precipitation at all on that day should be (1 - 0.3)^6 (that is, there is a 70% chance that no precipitation will fall for each of those given hours, so the chance that there will be no precipitation at all during that day is chance that there is no precipitation in hour 1 times the chance that there is no precipitation in hour 2 times the chance that there is no precipitation in hour 3, etc.). This yields a probability of zero precipitation of ~11.8%. Thus, the probability of getting precipitation at some point during the day is 1 - (1 - 0.3)^6 = 0.882351. Equivalently, there is a ~90% probability of precipitation at some point during the day.
Yet, whenever I compare daily forecasts with hourly forecasts, I see that the daily probability of precipitation never exceeds the maximum probability from the hourly probability of precipitation (i.e. to continue the previous example, the daily probability of precipitation is shown as 30%: the maximum of the hourly probabilities).
Is there some difference in how hourly and daily forecasts are calculated? What timescale truly represents the probability of receiving appreciable precipitation in a given area? What am I missing here?
I have no experience with meteorology, but it is one of the archetypal examples my math professors list as being a system that is incredibly difficult to predict, because there is no one leading feature (as in, there aren't any factors that provide a good indicator of the weather by themselves). In fact, I'm pretty sure most mathematicians use chaos theory to model weather. All of this is to say that a heck of lot more go into those hourly predictions than a product of two features (even though that may be a good intuitive starting point, and perhaps taking that product is the last step of a complicated process). My guess is that some form of machine learning or advanced statistical method is used for those predictions. And I do have experience with those two things.
I would assume that a different model is used for the hourly forecasts than the daily forecasts. Determining the weather per hour is a harder problem than the weather per day, so I imagine the hourly predictions are built upon the daily predictions. And you can bet that the way the hourly predictions are based on the daily predictions is not easily discernible.
Essentially, there's a lot of complicated, difficult math that goes into predictions like these, and they're going to be very hard to reproduce without being given the exact method used. It's like tasting a gourmet dish at a fancy restaurant, and trying to reproduce the exact recipe from one bite. Technically possible, but not probable.