This question is about quantifying information gained from weather forecast.
Suppose I know that the city I live in rains 30% of the time. Every night I listen to the weather forecast and try to guess whether it will rain the next day. Suppose the forecast is right 80% of the time, and in particular it's always right when it predicts rain (so it always rains when the forecast says it will). What is the best strategy I can adopt to guess whether it will rain, compared to if I didn't listen to the weather forecast and just had the knowledge that it rains 30% of the time? How do I quantify the amount of information I gain (in terms of entropy) from listening to the forecast and how do I compute the mutual information between my guess and the actual weather?
Edit: P(rain|forecast predicts rain)=1, but what is P(rain|forecast predicts no rain) if the overall accuracy of forecast is 80%? If I listen to the forecast, then I will guess rain whenever it predicts rain. How about if it predicts no rain? I guess the accuracy will still be higher than my 30%?