Suppose at this time of year in Auckland, there is a 50% chance of rain. Given that it rains, there is an 80% chance the forecaster will be correct in predicting "Rain". Given that it does not rain, there is 70% chance the forecaster will be correct in predicting "No Rain".

(a) The sample space for this problem is = {"rain" day and "no rain" day}. Define two events mentioned in the question.

(b) Using the events you defined in (a), write down the information given in terms of proba- bility statements.

(c) What is the probability that a randomly chosen day is a rainy day and the weather fore- cast predicts that it rains?

(d) If the weather forecast says it rains on a particular day, what is the probability that it actually rains?

(e) What is the probability that a randomly chosen day is either a rainy day or the weather forecast predicts that it rains?