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

I have now properly taken into account that when the economy is good, there is an 80% chance the study will predict good economy and a 20% chance it will predict poor economy. Similarly, when the economy is poor, there is a 10% chance the study will predict good economy and a 90% chance it will predict poor economy. I hope this helps! :)In summary, two states of nature exist for a particular situation: a good economy and a poor economy. A study can be conducted to determine the likelihood of either of these states occurring in the coming year. Currently, there is a 60% chance that the economy will be good and a 40% chance that it will be poor. Based on past data
  • #1
hildanhk
2
0
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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  • #2
Hi hildanhk,

Can you show us what you have tried so our helpers can see where you are stuck and can then offer help?:)
 
  • #3
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
 
  • #4
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:
  • #5
I have updated my previous post, since I made a mistake with my interpretation of the wording in the problem statement.
 

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

1. What is Bayes' theorem and why is it important?

Bayes' theorem is a mathematical formula used to calculate the probability of an event occurring based on prior knowledge or information. It is important because it allows us to update our beliefs or predictions as new evidence is presented.

2. How do you solve a Bayes' theorem problem?

To solve a Bayes' theorem problem, you need to identify the prior probability, likelihood, and evidence. Then, you can plug these values into the formula P(A|B) = P(B|A) * P(A) / P(B) and calculate the posterior probability.

3. What are some common misconceptions about Bayes' theorem?

Some common misconceptions about Bayes' theorem include thinking that it only applies to medical or legal fields, or that it is a subjective calculation. In reality, Bayes' theorem can be used in any field where we want to update our beliefs based on new evidence, and it is a purely mathematical calculation.

4. How can I improve my understanding of Bayes' theorem?

To improve your understanding of Bayes' theorem, you can practice solving problems and familiarize yourself with different applications of the theorem. You can also read up on the concepts and principles behind Bayes' theorem, such as conditional probability and prior knowledge.

5. What are some real-world examples of Bayes' theorem in action?

Bayes' theorem is commonly used in fields such as medicine, law, finance, and data science. For example, doctors may use it to calculate the probability of a disease based on a patient's symptoms and medical history. In finance, it can be used to predict stock market trends. In data science, it can be used for spam filtering or facial recognition technology.

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