A factory manufactures screws, machines X, Y and Z manufacture
respectively 1000,2000, 3000 of the screws,1%, 1.5% and 2 % of their
outputs are respectively defective .A screw is drawn at random from the
product and is found to be defective. What is the probability that it is
manufactured by the machine X ?
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Figure out how many defective screws are made by each machine. The probability that a defective screw is made by machine x is just the ratio of the number of defective screws made by machine x to the total number of defective screws.
there are total 1000 + 2000 +3000 screws = 6000
we chose a screw,P(it comes from X) = 1000/6000 = 1/6
P(it comes from B) = 1/3
P( " " " C) = 1/2
Now we know it is defective
P(defective from X) = 1% = 0.01
P(defective from Y) = 0.015
P( " " Z) = 0.02
Our favorable outcome is P(selecting from X)*P(defective from X)
total probability = P(selecting from X)*P(defective from X) + P(selecting from Y)*P(defective from Y) + P(selecting from Z)*P(defective from Z)
so required probability = fav probab/total probab
just substitute values.
NOTE - I used Baye's theorem.
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That's a complicated way of arriving at the same result :-) No need to know the percentages of selecting from each machine — you just need to know how many defective screws were made by each machine. Non-defective screws are irrelevant in this problem.
If we were trying to find the problem that a randomly selected screw would turn out to be a defective screw made by machine X, the extra work would be useful.