introduction to engineering mathematics volume 3 by h.k. dass

Question

Three machines I,II,III manufacture respectively 0.4,0.5,0.1 of the total production.The percentage of  defective items produced by I,II,III is 2,4,1 percent respectively.for an item chosen at random ,what is the probability that it is defective ?

Comments

What is the answer ?I got .029

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answer is 0.029

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why do we do not consider the probability of chosing a respective machine in this question as three machine are different ? defective item can be from any machine

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See my answer.We will select machine. Selecting A probability is .4 and defective from A is .01 so we get .4*.01 ,so this is the probability that item is defective and we select A.Now smilarly find other 2 and add all to get total probability.You can see 1-2 examples of Bayees theorem .

Answer 1

All the events are mutually exclusive and these are collective exhaustive,so its a direct application of Bayee's Theorem.

Defective can come from I or II or 3

So  0.4*.02 + .5*.04 + .1*.01 =.029