A furniture manufacturer sells three types of products: chairs, tables, and beds. Chairs constitute 45% of the company's sales, tables constitute 35% of the sales, and beds constitute the rest. Of the company's chairs, 8% are defective and have to be returned to the shop for minor repairs, whereas the percentage of such defective items for tables and beds are 4% and 5% respectively. A quality control manager just inspected an item and the item was not defective. What is the probability that this item was a chair?

Answer:The probability that this non defective product is a chair is 44.04 %.Step-by-step explanation:Given:  The probability of getting a product as chair is, [tex]P(A)=45\%=0.45[/tex]  The probability of getting a product as table is, [tex]P(B)=35\%=0.35[/tex]  The probability of getting a product as bed is, [tex]P(C)=20\%=0.20[/tex]  Now, let event D be having a defective product at random.  So, as per the question,  Probability of producing a defective product as chair is, [tex]P(D/A)=8\%=0.08[/tex]  Probability of producing a non defective product as chair is [tex]P(Not\ D/C)=100 - 8 = 92%=0.92[/tex] Probability of producing a defective product as table is, [tex]P(D/B)=4\%=0.04[/tex]  Probability of producing a defective product as bed is, [tex]P(D/C)=5\%=0.05[/tex]  Now, probability of having a defective product when selected at random is given as:  [tex]P(D)=P(A)\cdot P(D/A)+P(B)\cdot P(D/B)+P(C)\cdot P(D/C)\\P(D)=(0.45\times 0.08)+(0.35\times 0.04)+(0.20\times 0.05)\\ P(D)=0.036+0.014+0.01\\P(D)=0.06[/tex]Now, probability of selecting a non defective product is =  1 - 0.06 = 0.94Now, probability of selecting a product to be chair given that it is non defective is given using Bayes' Theorem and is given as:[tex]P(A/Not\ D)=\frac{P(A)\cdot P(Not\ D/A)}{P(Not\ D)}\\P(A/Not\ D)=\frac{0.45\times 0.92}{0.94}=\frac{0.414}{0.94}=0.4404[/tex]Therefore, the probability that this non defective product is a chair is 44.04 %