I am stuck on one of the questions for my class home work. I did the first question
A toy factory produces its toys with three machines A, B, and C. From the
total production, 50% are produced by machine A, 30% by machine B, and
20% by machine C. Past statistics show that 4% of the toys produced by
machine A are defective, 2% produced by machine B are defective, and 4%
of the toys produced by machine C are defective.
(a) What is the probability that a randomly selected toy is defective?
(b) If a randomly selected toy was found to be defective, what is the probability
that this toy was produced by machine A?
Let D be the event that the selected product is defective.
Then, Pr(A) =
0.5, Pr(B) = 0.3, Pr(C) = 0.2, Pr(D|A) = 0.04, Pr(D|B) = 0.02, Pr(D|C) =
0.04. We have
Pr(D) =Pr(D|A)Pr(A) + Pr(D|B)Pr(B) + Pr(D|C)Pr(C)
=(0.04)(0.50) + (0.02)(0.30) + (0.04)(0.20) = 0.034
(b) By Bayes’ theorem, we find
Pr(A|D) = Pr(D|A)Pr(A)
Pr(D)=
(0.04)(0.50)
0.034
≈ 0.5882
The second question is sort of the same but does not give me how much each company makes
Problem 2
A company is buying certain machine pieces from three different vendors.
Vendor A’s machine you get are broken 15% of times vendor B’s 20% and vendor C 30% of time. Overall the company receives broken machines 20% of time. An inspector randomly chooses one of the machines. What is a probability that the machine is broken and from vendor A?
so what i did was use 20% overall as the P(D) so
Pr(D|A)Pr(A)+ Pr(D|B)Pr(B)+ Pr(D|C)Pr(C) = (.15)(.20)+(.20)(.20)+(.30)(.20)=0.13
So Prob that the mchine is Brocken and from A is (Pr(D|A)Pr(A))/Pr(D)= (.15)(.20)/(0.13)≈ 0.23
IS THIS CORRECT OR AM I WRONG?
Best Answer
There is not enough information to solve the second problem. If it gets all its machine pieces from B, which is perfectly possible from the given numbers, then the probability the thing comes from A and is broken is $0$. But it also possible to arrange the buying pattern so that a non-zero proportion is bought from A, and there is a non-zero probability the machine piece is broken and comes from A.