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

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, 28% contain one defective component, and 21% 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 0.5898 one defective component 0.2591 two defective components 0.1506 (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 0 one defective component 0.0695 two defective components 0.9279

Transcript

00:01 So in this question we're told that 51 % of batches contain no defective components.

00:08 So the probability of no defective components is 0 .51.

00:14 Probability of one defective component is 0 .28.

00:18 And the probability of two defective components is 0 .21.

00:24 Now in the first kind of condition, neither tested component is defective.

00:34 So what's the probability of getting no defective components? so let's say in the test we get zero defective components.

00:49 Well this is going to be the probability of getting none given that the getting none and you're in the situation where you're batch has none, so that's 0 .51 times 1.

01:04 So let's say that you have one defective component in your batch, 0 .28, then your probability of getting no defective components in your test is going to be the probability that you pick two components and neither of them are defective.

01:25 So that's going to be 9 tenths times 8 9ths, because you pick the first component.

01:39 You have a 9 tenths chance of getting one that's not defective.

01:42 And when you pick a second component out, there's eight that aren't defective left and one that is.

01:48 And then let's say you have two in your batch that are defective.

01:53 Then what's your probability of not getting any of them? well, it's 8 out of 10 times 7 out of 9.

02:01 So let's say what is that then? so this is 0 .51 times 1 plus 0 .28 times 9 tenths times 8 9ths is 8 tenths, so 4 5ths, plus 0 .21 times 8 times 7 over 9 times 10.

02:26 And this is 1 -297 over 1 ,500.

02:31 So this is our probability of getting no components tested as defective.

02:40 And the reason we're calculating that is because the probability of having x components be defective, given that none test effective, is the probability of having x components that are defective and none of them test effective, divided by the total probability.

02:58 That none test defective.

03:02 So the probability of having no defective components given none of them test effective is the probability of getting no defective components and none of them test effective.

03:15 So that's 0 .51 times 1 divided by the total probability that none test effective, 1297 over 1 ,500.

03:28 So 0 .51 divided by that gives us 0 .5898.

03:40 So now the probability of having one defective component given that none test effective is the probability of having one defective component and not getting any tested defective.

03:52 So that's this proportion here.

03:54 0 .28 times 8 tenths.

03:56 So that's 0 .28 times 4 .5s divided by 1297 over 1 ,500.

04:04 So in 0 .28...

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