Question 1200376: In a suburban community, 30% of the households use Brand A toothpaste, 27% use Brand B, 25% use Brand C, and 18% use Brand D. In the foul groups of households, the proportions of residents who learned about the brand they use through television advertising are as follows Brand A, 0.10; Brand B, 0.05; Brand C, 0.20; and Brand D, 0.15. In a household selected at random from the community, it is found that residents learned about the toothpaste through television advertising. What is the probability that the brand of toothpaste used in the household is (a) A, (b) B, (c) C, (d)D?
Answer by GingerAle(43)   (Show Source):
You can put this solution on YOUR website! Certainly, let's calculate the probabilities.
**1. Define Probabilities**
* **Prior Probabilities (Brand Usage):**
* P(A) = 0.30 (Brand A)
* P(B) = 0.27 (Brand B)
* P(C) = 0.25 (Brand C)
* P(D) = 0.18 (Brand D)
* **Likelihoods (Learning through TV):**
* P(TV | A) = 0.10 (Probability of learning about Brand A through TV)
* P(TV | B) = 0.05 (Probability of learning about Brand B through TV)
* P(TV | C) = 0.20 (Probability of learning about Brand C through TV)
* P(TV | D) = 0.15 (Probability of learning about Brand D through TV)
**2. Calculate Joint Probabilities**
* Joint Probability = Prior Probability * Likelihood
* P(A and TV) = P(A) * P(TV | A) = 0.30 * 0.10 = 0.03
* P(B and TV) = P(B) * P(TV | B) = 0.27 * 0.05 = 0.0135
* P(C and TV) = P(C) * P(TV | C) = 0.25 * 0.20 = 0.05
* P(D and TV) = P(D) * P(TV | D) = 0.18 * 0.15 = 0.027
**3. Calculate Total Probability of Learning through TV**
* P(TV) = P(A and TV) + P(B and TV) + P(C and TV) + P(D and TV)
* P(TV) = 0.03 + 0.0135 + 0.05 + 0.027 = 0.1205
**4. Calculate Posterior Probabilities (Using Bayes' Theorem)**
* P(A | TV) = P(A and TV) / P(TV) = 0.03 / 0.1205 ≈ 0.2490
* P(B | TV) = P(B and TV) / P(TV) = 0.0135 / 0.1205 ≈ 0.1120
* P(C | TV) = P(C and TV) / P(TV) = 0.05 / 0.1205 ≈ 0.4149
* P(D | TV) = P(D and TV) / P(TV) = 0.027 / 0.1205 ≈ 0.2241
**Therefore:**
* (a) Probability the brand is A given TV advertising: **0.2490**
* (b) Probability the brand is B given TV advertising: **0.1120**
* (c) Probability the brand is C given TV advertising: **0.4149**
* (d) Probability the brand is D given TV advertising: **0.2241**
**In summary:**
* If a randomly selected household learned about their toothpaste through TV advertising, it is most likely that they use **Brand C** (with a probability of approximately 0.4149).
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