Question 344412
Your tree diagram would look something like this:


<pre>


                     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

</pre>


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.