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Coolwindows corp. produces 2,000 air conditioners per week in three plants: plant i, plant ii, and plant iii. plant i produces 800 air conditioners per week, plant ii produces 500 air conditioners per week, and plant iii produces 700 air conditioners per week. the probability of an air conditioner being defective is 7% for plant i, 5% for plant ii, and 6% for plant iii, respectively. in a random check, a defective air conditioner is found. what is the probability that the air conditioner was produced in plant ii

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mathmate
Knowledge on the definition and notation of conditional probability is also required.
P(A|B) means probability that event A is a success given that B has happened.
Definition: P(A|B)=P(AnB)/P(B).
"n" is used to mean ∩  or the intersection symbol in sets

Bayes theorem is the key to the solution.

Here we are given
Let d=event that a unit is defective
Total daily production = 2000
P(i)=800/2000=0.4      P(D|i)=0.07
P(ii)=500/2000=0.25   P(D|ii)=0.05
P(iii)=700/2000=0.35  P(D|iii)=0.06
[total =sum P()=1.0]
We need to find P(ii|D), i.e. probability that unit is produced from plant ii given that it is defective.

overall probability of a defective unit
P(D)=P(D|i)*P(i) + P(D|ii)*P(ii) + P(D|iii)*P(iii)  [law of total probability]
=0.4*0.07+0.25*0.05+0.35*0.06
=0.0615

Apply Baye's theorem:
P(ii|D)=P(ii n D)/P(D)    [ definition of conditional probability]
=P(D|ii)*P(ii) / P(D)       [ application of Baye's theorem, or def. of cond. prob.]
=0.05*0.25/0.0614
=0.2036

Comment on result:
The proportion of units produced in ii is 0.25.  However, the defective rate is the lowest of the three plants, hence it is reasonable that the probability that ii produced the defective unit is lower than 0.25, like 0.2036.
Logical glance like this gives a quick check of answers.

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