Components of a certain type are shipped to a supplier in batches of ten. Suppose that 50% of all such batches contain no defective components, 30% contain one defective component, and 20% contain two defective components. Two components from a batch are randomly selected and tested. What are the probabilities associated with 0, 1, and 2 defective components being in the batch under each of the following conditions? a.? ?Neither tested component is defective. b. ?One of the two tested components is defective. [?Hint: ?Draw a tree diagram with three first-generation branches for the three different types of batches.]

Answer Step 1 of 5 a) Let 0 be the event that the batch has 0 defectives 1 be the event the batch has 1 defective, 2 be the event the batch has 2 defectives. Let 0 be the event that neither selected component is defective. (0 ) = 0.5, (1 ) = 0.3, (2 ) = 0.2 Step 2 of 5 The event 0 can happen in three different ways: (i) Our batch of 10 is perfect, and we get no defectives in our sample of two; P(D /B )=0.5(1)=0.5 0 0 (ii) Our batch of 10 has 1 defective, but our sample of two misses them; P(D /B )=0.3 9 c =0.24 0 1 10 c (iii) Our batch has 2 defective, but our sample misses them. P(D /B )=0.2 8c2 =0.124 0 2 10 c P(D 0= P(D /B0)+0(D /B 0 P1 /B ) 0 2 =0.5+0.24+0.124=0.864 Step 3 of 5 The probabilities associated with 0, 1, and 2 defective components neither tested component is defective. P(B /D )= P(D0/B0) =0.5/0.864=0.5787 0 0 P(D0) P(D0/B1) P(B 1 )0 P(D0) =0.24/0.864=0.278 P(D0/B2) P(B 2 )0 P(D0) =0.124/0.864=0.144