Question

companies a, b, and c produce 10%, 10%, and 80%, respectively, of the major appliances sold in a certain area. in that area, 4% of the company a appliances, 4\\(\frac{1}{2}\\)% of the company b appliances, and 4% of the company c appliances need service within the first year. suppose a defective appliance is chosen at random; find the probability that it was manufactured by company b.\
the probability that it came from company b is \\(\square\\).\
(type an integer or decimal rounded to four decimal places as needed)

Explanation:

Step1: Define given probabilities

Let $P(A)=0.10$, $P(B)=0.10$, $P(C)=0.80$ (market shares).
Let $P(D|A)=0.04$, $P(D|B)=0.045$, $P(D|C)=0.04$ (defect rates, where $4\frac{1}{2}\%=0.045$).

Step2: Calculate total defect probability

Use law of total probability:

$$\begin{align*} P(D)&=P(D|A)P(A)+P(D|B)P(B)+P(D|C)P(C)\\ &=(0.04\times0.10)+(0.045\times0.10)+(0.04\times0.80)\\ &=0.004+0.0045+0.032\\ &=0.0405 \end{align*}$$

Step3: Apply Bayes' Theorem

Calculate $P(B|D)$:

$$ P(B|D)=\frac{P(D|B)P(B)}{P(D)}=\frac{0.045\times0.10}{0.0405} $$

Answer:

0.1111