In a certain country, it rains 40% of the days and shines 60% of the days. A barometer manufacturer, in testing his instrument in the lab, has found That it sometimes errs: on rainy days it erroneously predicts "shine" il0 % of the time, and on shiny days it erroneously predicts "rain" 30% of the time.

(a) In predicting tomorrows weather before looking at the barometer, the (prior) chance of rain is 40%. After looking at the barometer and seeing it: predict "rain," what is the (posterior) chance of rain?

(b) What is the posterior chance of rain if an improved barometer (error rates of 10 and 20% respectively) predicts "rain"?

(c) What is the posterior chance of shine if the improved barometer predicts rain" ?