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.