Question 1171673: ⦁ A marketing manager believes the market demand potential of a new product to be high with a probability of 0.30, or average with probability of 0.50, or to be low with a probability of 0.20. From a sample of 20 employees, 14 indicated a very favorable reception to the new product. In the past such an employee response (14 out of 20 favorable) has occurred with the following probabilities: if the actual demand is high, the probability of favorable reception is 0.80; if the actual demand is average, the probability of favorable reception is 0.55; and if the actual demand is low, the probability of the favorable reception is 0.30. Thus given a favorable reception, what is the probability of actual high demand? (5 marks)
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! To solve this problem, we'll use Bayes' Theorem. Here's how we break it down:
**1. Define the Events:**
* H: High demand
* A: Average demand
* L: Low demand
* F: Favorable reception (14 out of 20 employees)
**2. Given Probabilities:**
* P(H) = 0.30 (Prior probability of high demand)
* P(A) = 0.50 (Prior probability of average demand)
* P(L) = 0.20 (Prior probability of low demand)
* P(F|H) = 0.80 (Probability of favorable reception given high demand)
* P(F|A) = 0.55 (Probability of favorable reception given average demand)
* P(F|L) = 0.30 (Probability of favorable reception given low demand)
**3. Calculate the Marginal Probability of Favorable Reception (P(F)):**
* P(F) = P(F|H)P(H) + P(F|A)P(A) + P(F|L)P(L)
* P(F) = (0.80 * 0.30) + (0.55 * 0.50) + (0.30 * 0.20)
* P(F) = 0.24 + 0.275 + 0.06
* P(F) = 0.575
**4. Apply Bayes' Theorem to Find P(H|F):**
* P(H|F) = [P(F|H) * P(H)] / P(F)
* P(H|F) = (0.80 * 0.30) / 0.575
* P(H|F) = 0.24 / 0.575
* P(H|F) ≈ 0.41739
**5. Round to Two Decimal Places:**
* P(H|F) ≈ 0.42
**Therefore, the probability of actual high demand given a favorable reception is approximately 0.42.**
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