Weather station brain teaser I am living in a world where tomorrow will either rain or not rain. There are two independent weather stations (A,B) that can predict the chance of raining tomorrow with equal probability 3/5. They both say it will rain, what is the probability of it actually rain tomorrow?
My intuition is to look at the complementary, i.e. $$1-P(\text{not rain | A = rain and B = rain}) = \frac{21}{25}$$
However, using the same methodology, the chance of it not raining tomorrow is: $$1-P(\text{rain | A = rain and B = rain}) = \frac{16}{25}$$
Clearly they do not add up to 1!
Edit: corrected the probability to 3/5
** Edit 2:
There seems to be a problem using this method. Say the probability of getting a right prediction is 1/2, so basically there is no predictability power. The using the original argument, the probability of raining is $1-\frac{1}{2}*\frac{1}{2} = \frac{3}{4}$ which is also non-sensical

After a couple months of thinking, a friend of mine have pointed out that the question lacks one piece of information: the unconditional probability distribution of rain $P(rain)$. The logic is that, if one is to live in an area that is certain to rain everyday, the probability of raining is always 1. Then Kaz's analysis will give the wrong answer (9/13). In fact, the probability of correctly predicting the rain should be:
$$\frac{P(rain)(3/5)^2}{P(rain)(3/5)^2+(1-P(rain))(2/5)^2}$$
Kaz's answer is correct if the probability distribution of raining is uniform. Cheers.
