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        riddles >> medium >> Weather forecast
        
(Message started by: Altamira_64 on Apr 18^th, 2016, 11:26am)

Title: Weather forecast
Post by Altamira_64 on Apr 18^th, 2016, 11:26am

There are two weather stations, station A and station B which are independent of each other. On average, the weather forecast accuracy of station A is 80% and that of station B is 90%. Station A predicts that tomorrow will be sunny, whereas station B predicts rain. What is the probability that it rains tomorrow? We are not asking for the exact probability; we are just asking whether it is more likely to rain or not.

Title: Re: Weather forecast
Post by towr on Apr 20^th, 2016, 10:32am

[hide]I don't think there's any way to tell.
Suppose it's always raining, and A randomly predicts it's sunny 20% of the time, and B that it's sunny 10% of the time. There's 100% of rain tomorrow
Suppose it's always sunny, and A randomly predicts it's rainy 20% of the time, and B that it's rainy 10% of the time. There's 100% of sun tomorrow[/hide]

Title: Re: Weather forecast
Post by Grimbal on Apr 22^nd, 2016, 4:43am

Exactly my thought.
Not only the a-priori distribution is missing, but there is the question whether false positive and false negatives are eqially likely. And any correlation between A and B's predictions would probably change the game.

Title: Re: Weather forecast
Post by dudiobugtron on May 3^rd, 2016, 6:29pm

The stations are independent of each other. (And so, we assume, are their predictions.) This means that they are (effectively) making their predictions randomly, according to some probability distribution. The worries about correlations are dealt with by this independence.

The a-priori distribution is indeed missing, but why should that stop us? Can't we just work it out in the general case?

The biggest issue IMO is whether 'sunny' and 'raining' are mutually exclusive events.

Title: Re: Weather forecast
Post by Altamira_64 on May 5^th, 2016, 3:23am

You are right, I can make it explicitly exclusive, as follows: A predicts that it will NOT rain, while B predicts rain.

on 05/03/16 at 18:29:17, dudiobugtron wrote:

Title: Re: Weather forecast
Post by Grimbal on May 10^th, 2016, 1:29am

A and B's predictions are correlated with the actual weather. They can't be independent. You have to consier A and B's errors are independent. For example A and B make a correct prediction but randomly and independently change the prediction with 20% and 10% probability. I am not sure it is the only possible model.

Title: Re: Weather forecast
Post by dudiobugtron on May 10^th, 2016, 6:01pm

on 05/10/16 at 01:29:23, Grimbal wrote:

If they randomly change their prediction as you describe, this would be the same as making an incorrect prediction, then randomly and independently changing it to correct with 80% and 90% probability. If the errors are independent, then the correct predictions are too.

Independent means P(A) * P(B) = P(A and B)
In this example, if would mean P(A and B) = 72%
So there can still be a correlation , but it's not an issue as long as the two are independent.

Title: Re: Weather forecast
Post by Altamira_64 on May 11^th, 2016, 1:38am

Since the "a priori" probability is missing, we must assume it is 50% (thus it is equally likely to rain or not to rain) and we are asking for the probability, basis only in the two weather stations prediction.

Title: Re: Weather forecast
Post by Grimbal on May 13^th, 2016, 8:14am

@dudiobugtron I see.

I was referring to the correlation between the predictions, while the 80% and 90% obviously refer to the probability of being right.

Title: Re: Weather forecast
Post by riddler358 on May 24^th, 2016, 5:15am

by the simple naive approach

assuming events are exclusive and last entire day

i would say station A is twice more likely to make incorrect prediction, therefor it's 66,(6)% it will rain, and 33,(3)% it will be sunny