Probability of rain this weeken?

[TSN79]

Probability of rain this weeken??

If there is a 50% chance of rain on saturday, and the same on sunday, what's the probability of rain this weekend? The answer apparently is 75%, because three of four scenarios have rain. Following this theory, wouldn't you get the same answer if you asked "what's the probability of NO rain this weekend" ?

 

[Simon Bridge]

Almost.

The only way you'd say "there was no rain this weekend" would be if it was fine all weekend.

This is why statistitians are usually much more precise in their language, vis:
There is a 0.75 probability that there is rain on at least one day this weekend
There is a 0.75 probability that there is at least one fine day this weekend but only 0.25 probability that it is fine all weekend long.

Similarly, there is only 0.25 prob that it will rain both days.

Of course, this calculation assumes that the rainfall events are independent.

 

[SW VandeCarr]

Almost.

Of course, this calculation assumes that the rainfall events are independent.

Yes. Treating weather events closely related in time as independent events is inappropriate. Lacking a fully deterministic theory for such events, correlations are determined from long term data.

 

[Simon Bridge]

John Paulos (Innumeracy) uses the problem as an example... there it is represented as a reaction to a weather report: there's a 50% chance of rain on Saturday and a 50% chance of rain on Sunday, so there is a 100% chance of rain this weekend.

But what did the meteorologist mean? Perhaps there was, indeed, going to be rain in the weekend with equal chances of falling on either day? This, or something like it, may well have been the case if that worthy had indeed taken into account long-term data. So perhaps John shouldn't have scoffed?

 

[SW VandeCarr]

John Paulos (Innumeracy) uses the problem as an example... there it is represented as a reaction to a weather report: there's a 50% chance of rain on Saturday and a 50% chance of rain on Sunday, so there is a 100% chance of rain this weekend.

But what did the meteorologist mean? Perhaps there was, indeed, going to be rain in the weekend with equal chances of falling on either day? This, or something like it, may well have been the case if that worthy had indeed taken into account long-term data. So perhaps John shouldn't have scoffed?

This is just ignorance of probability on Paulos's part. If the events were independent, the probability of rain on the weekend is the sum for two independent events:

0.5 + 0.5 - (0.5)(0.5) = 0.75

Also, it's not just long term data involved in predictions. There are good theories about the behavior of weather systems, but predictions always carry a degree of uncertainty.

 

[Simon Bridge]

This is just ignorance of probability on Paulos's part.

Somehow I doubt that :)

John Allen Paulos (born July 4, 1945) is a professor of mathematics at Temple University in Philadelphia who has gained fame as a writer and speaker on mathematics and the importance of mathematical literacy. His book Innumeracy: Mathematical Illiteracy and its Consequences (1988) was an influential bestseller and A Mathematician Reads the Newspaper (1995) extended the critique.

http://en.wikipedia.org/wiki/John_Allen_Paulos
... he was using as an example of the sort of mistake he sees often.