MHB Bayes' theorem problem, Struggling with this the whole night, .Thank you.

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The discussion revolves around applying Bayes' theorem to determine the probabilities of a good or poor economy based on predictions from an economic study. Initially, the probabilities are set at 60% for a good economy and 40% for a poor economy, with the study's accuracy rates provided. A participant struggles with calculations, mistakenly using incorrect values in the formula, leading to an incorrect result of 0.51 instead of the correct 0.923 for the probability of a good economy given a prediction of good. Clarifications are offered regarding the correct interpretation of the probabilities and the formula application. The conversation emphasizes the importance of accurately applying Bayes' theorem to derive the correct posterior probabilities.
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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 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.
 
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Hi hildanhk,

Can you show us what you have tried so our helpers can see where you are stuck and can then offer help?:)
 
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
 
hildanhk said:
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:
I have updated my previous post, since I made a mistake with my interpretation of the wording in the problem statement.
 
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