A factory has three machines A,B and C, which produce 100,200 and 300 items of a particular type daily. The machines produce 2%, 3% and 5% defective items respectively. One day when the production was over, and item was picked up randomly and it was found to be defective. Find the probability that it was produced by machine A.
Given:
Machine A produce 100 items
Machine B produce 200 items
Machine C produce 300 items
Let us assume U1, U2, U3 and A be the events as follows:
U1 = choosing Machine A
U2 = choosing Machine B
U3 = choosing Machine C
A = Producing defective item by machine
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From the problem
⇒ P(A|U1) = P(Producing defective item by Machine A)
⇒ 
⇒ P(A|U2) = P(Producing defective item by Machine B)
⇒ 
⇒ P(A|U3) = P(Producing defective item by Machine C)
⇒ 
Now we find
P(U1|A) = P(The chosen defective item is produced by machine A)
Using Baye’s theorem:
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⇒ 
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∴ The required probability is 
.