Question 1198454: Market Researchers, Inc, has been hired to perform a study to determine it the market for a naw product will be good or poor. In similar studies performed in the past, whenever the market actunlly was good, the market research study indicated that it would be good 85% of the time On the other hand, whenever the market actually was poor. the market study incorrectly predicted it would be good 20% of the time. Before the study Is performed, It is believed there is a 70% chance the market will be good. When Market Researchers, Inc. performs the study for this product, the results predict the market will be good. Given the results of this study, what is the probability that the market actually will be good?
Answer by onyulee(41)   (Show Source):
You can put this solution on YOUR website! Certainly, let's calculate the probability that the market will actually be good given the market research study predicts it will be good.
**1. Define Events**
* **G:** Event that the market is actually good.
* **B:** Event that the market research study predicts the market will be good.
**2. Given Probabilities**
* P(G) = 0.70 (Probability that the market will be good)
* P(G') = 1 - P(G) = 0.30 (Probability that the market will be poor)
* P(B|G) = 0.85 (Probability that the study predicts good when the market is good)
* P(B|G') = 0.20 (Probability that the study predicts good when the market is poor)
**3. Calculate the Probability of the Study Predicting Good (P(B))**
* P(B) = P(B|G) * P(G) + P(B|G') * P(G')
* P(B) = (0.85 * 0.70) + (0.20 * 0.30)
* P(B) = 0.595 + 0.06
* P(B) = 0.655
**4. Calculate the Probability that the Market is Good Given the Study Predicts Good (P(G|B))**
* Use Bayes' Theorem:
P(G|B) = [P(B|G) * P(G)] / P(B)
P(G|B) = (0.85 * 0.70) / 0.655
P(G|B) = 0.595 / 0.655
P(G|B) ≈ 0.9084
**Therefore, the probability that the market will actually be good given that the market research study predicts it will be good is approximately 0.9084 or 90.84%.**
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