Clothing company

A clothing company manufactures three kinds of coats. The weekly production is 400 of the first type, 370 of the second, and 230 of the third type. By long-term observation it was found that among the quality coats there are 2/3 of the first type, 7/9 of the second type, and 4/5 of the third type.

What is the probability that:
a) a coat will be defective if we randomly select one in inspection;
b) with what probability may a randomly selected defective coat belong to the second type of coats.

Final Answer:

p₁ =  0.2616
p₂ =  0.3144

Step-by-step explanation:

$$\begin{gathered} a=\frac{2}{3}\doteq 0.6667 \\ b=\frac{7}{9}\doteq 0.7778 \\ c=\frac{4}{5}=0.8 \\ k=400\cdot a+370\cdot b+230\cdot c=400\cdot 0.6667+370\cdot 0.7778+230\cdot 0.8=\frac{6646}{9}=738\frac{4}{9}\doteq 738.4444 \\ s=400+370+230=1000 \\ {p}_1=(s-k)/s=(1000-738.4444)/1000=0.2616 \end{gathered}$$

$$\begin{gathered} n_1=400\cdot (1-a)=400\cdot (1-0.6667)=\frac{400}{3}=133\frac{1}{3}\doteq 133.3333 \\ {n}_2=370\cdot (1-b)=370\cdot (1-0.7778)=\frac{740}{9}=82\frac{2}{9}\doteq 82.2222 \\ {n}_3=230\cdot (1-c)=230\cdot (1-0.8)=46 \\ {p}_2=\frac{n_2}{n_1+n_2+n_3}=\frac{82.2222}{133.3333+82.2222+46}=0.3144 \end{gathered}$$

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