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Two states of nature exist for a particular situation: a good economy and a poor economy. An economic study may be performed to obtain more information about which of these will actually occur in the coming year. The study may forecast either a good economy or a poor economy. Currently there is a 50-50 chance that the economy will be good. In the past, whenever the economy is good, the economic study predicted it would be good 80% of the time. The other 20% of the time the prediction was wrong. In the past, whenever the economy is poor, the economic study predicted it would be poor 90% of the time. The other 10% of the time the prediction was wrong.
a. What is the probability of having a good economy if it is predicted it would be good?
b. What is the probability of having a good economy if it is predicted it would be poor?
c. Suppose the initial probability of good economy is 70% and the probability of a poor economy is 30%. Find the posterior probabilities in (a) using these new values.
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