Question 344412: Three machines A, B, and C are used to produce the same part, and thier outputs are collected in a single bin. Machine A produces 26% of the parts in the bin, B, 38% and machine C the rest. Of the parts produced by machine A, 8% are defective. Similary 5% of the parts from B and 4% of the parts from C are defective. Now how do you draw a probability tree to depict this situation.
B. If a part is chosen randomly from the bin, if a part is defective, what is the probability that it was produced by machine A, and if the part is good, what is the probability that it was produced by machine B.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! Your tree diagram would look something like this:
A B C
x x x
x x x
x x x
x x x
=.26*T =.38*T =.36*T
xxxxxxxxx xxxxxxxxx xxxxxxxxx
x x x x x x
x x x x x x
x x x x x x
x x x x x x
x x x x x x
x x x x x x
x x x x x x
=.08*A =.92*A =.05*B =.95*B =.04*C =.96*C
T = total number of parts in the bin.
A = total number of parts contributed by machine A.
B = total number of parts contributed by machine B.
C = total number of parts contributed by machine C.
Since A = .26 * T and B = .38 * T and C = .36 * T, then:
Total defective in the bin from machine A would be .26 * .08 = .0208 * T
Total defective in the bin from machine B would be .38 * .05 = .0190 * T
Total defective in the bin from machine C would be .36 * .04 = .0144 * T
Total defective parts in the bin is equal to (.0208 + .0190 + .0144) * T.
Total defective parts in the bin is therefore equal to .0542 * T.
Similarly,
Total good in the bin from machine A would be .26 * .92 = .2392 * T
Total good in the bin from machine B would be .38 * .95 = .3610 * T
Total good in the bin from machine C would be .36 * .96 = .3456 * T
Total good parts in the bin is equal to (.2392 + .3610 + .3456) * T.
Total good parts in the bin is therefore equal to .9458 * T.
The total good and bad in the bin should be equal to T.
.0542 * T + .9458 * T = 1 * T = T so this is ok.
Given that you pick a defective part out of the bin, what is the probability that it came from machine A?
The probability that it came from machine A would be .0208 / .0542 = .383763838.
Given that you pick a good part out of the bin, what is the probability that it came from machine B?
The probability that it came from machine B would be .3610 / .9458 = .38168746.
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