Question 1174231: A company is considering producing some new products. Based on past records, management believes that there is a 70 percent chance that each of these will be successful, and a 30 percent chance of failure. Market research may be used to revise these probabilities. In the past, the successful products were predicted to be successful based on market research 90 percent of the time. However, for products that failed, the market research predicted these would be successes 20 percent of the time. If market research is performed for a new product, what is the probability that the results indicate a successful market for the product and the product actually is successful?
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Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem using probability and Bayes' Theorem.
**Define Events:**
* **S:** The product is successful.
* **F:** The product fails.
* **M:** Market research indicates success.
**Given Probabilities:**
* P(S) = 0.70 (Probability of success)
* P(F) = 0.30 (Probability of failure)
* P(M|S) = 0.90 (Probability of market research indicating success given the product is successful)
* P(M|F) = 0.20 (Probability of market research indicating success given the product fails)
**What We Need to Find:**
* P(M and S) (Probability that market research indicates success AND the product is successful)
**Steps:**
1. **Use the conditional probability formula:**
* P(M|S) = P(M and S) / P(S)
2. **Rearrange the formula to solve for P(M and S):**
* P(M and S) = P(M|S) * P(S)
3. **Substitute the given probabilities:**
* P(M and S) = 0.90 * 0.70
4. **Calculate the result:**
* P(M and S) = 0.63
**Therefore, the probability that the market research results indicate a successful market for the product and the product actually is successful is 0.63 or 63%.**
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