A certain item is manufactured by 3 factory F₁, F₂ and F₃ with 30 % of item made in F₁, 20 % in F₂ and 50 % in F₃ . It is found that 2 % of the items produced by F₁ .3 % of the items produced by F₂ and 4 % of the items produced by F₃ are defective. Suppose that an items selected at random from the stock is found defective. What is the probability that the item came from F₁ ?

Step-by-step solution

Let events T₁, T₂, T₃ be the following T₁ : The item is manufactured by factory F₁ . T₂ : The item is manufactured by factory F₂ . T₃ : : The item is manufactured by factory F₃ . Clearly, T₁, T₂, T₃ are mutually exclusive and exhaustive events. aligned &P (T₁ )=30 %=0.3, P (T₂ )=20 %=0.2 &P (T₃ )=50 %=0.5 aligned Let E be the event that item is defective. Now, array ll & P (E T₁ )=2 %=0.02 & P (E T₂ )=3 %=0.03 & P (E T₃ )=4 %=0.04 array Hence, by Bayes' theorem, we have aligned P (T₁ E ) &= P (T₁ ) P (E T₁ ) P (T₁