Question

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, 33% contain one defective component, and 16% 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.

no defective components :

one defective component :

two defective components :

(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.]

no defective components :

one defective component :

two defective components :

Answer 1

Let be the event that the batch has 0 defectives, be the event the batch has 1 defective, and be the event the batch has 2 defectives. Let be the event that neither selected component is defective.

The event can happen in three different ways: (i) Our batch of 10 is perfect, and we get no defectives in our sample of two; (ii) Our batch of 10 has 1 defective, but our sample of two misses them; (iii) Our batch has 2 defective, but our sample misses them

For i) the probability is (0.5)(1)

for ii), the probability that our batch has 1 defective is 0.33. Given that it has 1 defective, the probability that our sample misses it is

So the probability of (ii) is

For (iii), the probability our batch has 2 defective is 0.16.

Given that it has 2 defectives, the probability that our sample misses both is . So the probability of (iii) is . We have therefore found that