Components of a certain type are shipped to a supplier in ba

Components of a certain type are shipped to a supplier in batches of ten. Suppose that 51% of all such batches contain no defective components, 30% contain one defective component, and 19% 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? (Round your answers to four decimal places.)

(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.]

Solution

Let E1,E2 and E3 denote the events that a two components are selected at random associated with no defective, 1 defective and 2 defective respectively , then P(E1)=0.51,P(E2)=0.3 & P(E3)=0.19.

(b): let E denote the event of two tested components is defective,P(E/E1)=0.3, P(E/E2)=0.1 & P(E/E3)=0. Then

P(E1/E)=P(No defective)=(0.51*0.3)/[(0.51*0.3)+(0.3*0.1)+(0.19*0)]=0.3377.

P(E2/E)=P(one defective)=(0.1*0.3)/[(0.51*0.3)+(0.3*0.1)+(0.19*0)]=0.6622.

P(E3/E)=P(two defective)=(0.19*0)/[(0.51*0.3)+(0.3*0.1)+(0.19*0)]=0.