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hildanhk

New member
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 60% chance that the economy will be good and a 40% chance that the economy will be poor. In the past, whenever 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 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.

anemone

MHB POTW Director
Staff member
Hi hildanhk,

Can you show us what you have tried so our helpers can see where you are stuck and can then offer help? hildanhk

New member
I am try to use the formula and plug the numbers in. But I could not get the correct answers.
P(A|B)= P(B|A)P(A)/P(B|A)P(A)+P(B|A')P(A')

For example, I put
P(good economy| prediction of good economy)= 0.8*0.6/0.8*0.6+0.9*0.6= 0.51

But the answer should be 0.923

Klaas van Aarsen

MHB Seeker
Staff member
I am try to use the formula and plug the numbers in. But I could not get the correct answers.
P(A|B)= P(B|A)P(A)/P(B|A)P(A)+P(B|A')P(A')

For example, I put
P(good economy| prediction of good economy)= 0.8*0.6/0.8*0.6+0.9*0.6= 0.51

But the answer should be 0.923
Welcome to MHB, hildanhk! You seem to have mixed up the last part (0.9 * 0.6).

EDIT: The phrasing is somewhat confusing.
We have P(predict good | good) = 80%.
The remainder is P(predict poor | good) = 20%
And from P(predict poor | poor) = 90%,
we get that P(predict good | poor) = 10%.

So it should be (edited):
$$P(B|A')\ P(A') = P(\text{predict good}|\text{poor})\ P(\text{poor}) = 0.1 \cdot 0.4$$

So you should have:
$$P(\text{good}|\text{predict good}) = \frac{0.8 \cdot 0.6}{0.8 \cdot 0.6 + 0.1 \cdot 0.4} \approx 0.923$$

Last edited:

Klaas van Aarsen

MHB Seeker
Staff member
I have updated my previous post, since I made a mistake with my interpretation of the wording in the problem statement.