Question

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 60% chance that the economy will be good and a 40% chance that it will be poor. In the past, when ever the economy was 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 was poor, the economic study predicted it would be poor 90% of the time. (The other 10% of the time the prediction was wrong.)

(A) Use Bayes’ theorem and find the following:

P (good economy | prediction of good economy)

P (poor economy | prediction of good economy)

P (good economy | prediction of poor economy)

P (poor economy | prediction of poor economy)

(B) Suppose the initial (prior) probability of a good economy is 70% (instead of 60%), and the probability of a poor economy is 30% (instead of 40%). Find the posterior probabilities in part a

based on these new values.

Answer 1

Probability of good economy, P(good) = 0.60

Probability of poor economy, P(poor) = 0.40

Probability of prediction of good economy given that economy was good, P(PG|good) = 0.8

Probability of prediction of poor economy given that economy was good, P(PP|good) = 0.2

Probability of prediction of poor economy given that economy was poor, P(PP|poor) = 0.9

Probability of prediction of good economy given that economy was poor, P(PG|poor) = 0.1

a) i) P(good economy | prediction of good economy) =

ii) P(poor economy | prediction of good economy) = 1 - P(good economy | prediction of good economy)

iii) P(good economy | prediction of poor economy)

iv) P(poor economy | prediction of poor economy) = 1 -  P(good economy | prediction of poor economy)

b) If P(good) = 0.7

and P(poor) = 0.3

i) P(good economy | prediction of good economy) =

ii) P(poor economy | prediction of good economy) = 1 - P(good economy | prediction of good economy)

iii) P(good economy | prediction of poor economy)

iv) P(poor economy | prediction of poor economy) = 1 - P(good economy | prediction of poor economy)