Contributed by Anders Persson
As you may know, one of the more lively and inventive weather services in Europe is the regional meteorological office in Bologna. They have not only developed a good limited area model (COSMO) but also an ensemble system (COSMO LEPS) to complement it.
They have also, like me, come to the conclusion that using probabilistic systems demands some knowledge and feeling for probabilities. For that purpose they are arranging a course in February “Probability and uncertainty: two concepts to be expanded in meteorology”.
I have the honour to be one of the invited lecturers and presently I am busy preparing my presentations, drawing partly on previous seminars and lectures. Doing so, I am sometimes confronted with the problem to understand myself or things I have said previously! Here some examples I would like to share with you.
The sack with marbles
In a seminar in 2005, I seem to have said, en passant, something like this:
“Consider a sack with a large number of bland and white marbles in unknown proportions. Draw one. Then draw another one. The probability that the second has the same colour as the first is 2/3.”
This sounds ridiculous. How could I have said that? There was no explanation and I remember only vaguely that I had seen it in a book, now forgotten.
It is elementary knowledge that if there are 80% (0.8) black marbles the chance of drawing two identical is 64% (0.82=0.64) and 4% for both white 0.22=0.04). And if there were 30% (0.3) blacks the chance would be 9% (0.32=0.09) and 49% for both white marbles (0.72=0.49).
But then it struck me that these were answers to questions never raised. It was not about black or white, but about both colours without specifying which was more or less numerous. And 64% + 4% = 68% and 49% + 9% = 58% are both not far from 2/3 = 67%.
Then I realised that, if P is the proportion of black marbles and 1-P the proportion of white marbles, the chance of getting identical colours is :
P2 + (1-P)2 = 2P2 – 2P + 1.
If you integrate you’ll have:
2P3/3 – P2 + P + const
Which, evaluated from 0% to 100%, yields 2/3!!
Encouraged by this, I posed the question: – If both marbles drawn from the sack are black, what does that tell us about the likely proportions between black and white marbles in the sack?
This leads into very interesting calculations which are sometimes called “Bayesian” which I will not reveal here, but you are welcome to Bologna to find out! Or wait until my next blog post!
But there is another headache I would like to share with you… and where I do not have the answers!
The 1930’s Californian weather service
To illustrate the advantage of probabilistic weather forecasts compared to deterministic forecasts, I used in 2005 an example from the 1930’s dry and sunny California.
A newly created private weather service earned loads of $$$$$ supplying its customers with highly biased rain forecasts. Sounds stupid, but these customers were the Hollywood movie industry and the Californian water industry. The former didn’t want to risk huge costs by being surprised by rain when they were shooting outdoor so they got forecasts highly over-estimating the frequencies of rain. They didn’t mind false rain alarms. The water industry (water dams for electric power and irrigation) got forecasts which rarely mentioned rain unless it was pretty certain to happen. They didn’t mind missed rain events.
In 2005, I had explained this by their interest to avoid over-filling and possible damages to the dam by spilling off water. But this would be done unnecessarily if there was no rain. But ten years later, 2015, I do not quite understand this explanation. Because if the dam is almost full and you risk over-filling, then of course you would be equally concerned with non-predicted rain. But they were not.
Perhaps it was not about the water volumes as such, but the commercial side? When it was raining the irrigation people didn’t need to buy water, unless the price was radically lowered. Knowing when rain was absolutely sure the weather authorities could in advance offer those cheaper prices. And then of course if there was no rain, they would have lowered the prices unnecessarily.
But I am not sure this is true. I know very little about water economy and nothing about the hydrological culture in California, in particular in the 1930’s.
So is there anybody out there who does? Or who can give some other, perhaps completely different explanation to why weather authorities (in any part of the world at any time) would favour under-prediction of rain in this way?