Acceptance Sampling Suppose that you have just received a ship...

Question

Acceptance Sampling Suppose that you have just received a shipment of 100 televisions. Although you don’t know this, 6 are defective. To determine whether you will accept the shipment, you randomly select 5 televisions and test them. If all 5 televisions work, you accept the shipment; otherwise, the shipment is rejected. What is the probability of accepting the shipment?

Accepted Answer

Welcome back, everyone. In this problem, a shipment contains 80 laptops, of which 5 are defective, unknown to the inspector. The inspector randomly tests 4 laptops without replacement. The shipment is accepted only if none of the 4 tested laptops is defective. What is the probability of accepting the shipment? A says it's 0.0585, B 0.7685, C 0.0089, and D 0.5180. Now to solve this problem, we'll need the help of the concept of combinations and probability for drawing objects without replacement because remember if we're trying to find the probability, we'll need to find the total number of outcomes, that is all the ways that we could choose 54 laptops from 80, sorry, and we'll need to figure out the number of favorable outcomes, that is selecting only good laptops. Now what do we know about combinations? Well, recall that for a combination and choose R, it's going to be equal to. In factorial divided by R factorial multiplied by N minus R factorial, where N is the total number of laptops in this case and R is the number of laptops being selected. Now for this problem. The total number of ways, let's write that here. So this this is the total number of outcomes. The total number of ways. To select 4 laptops, so are you call 4 laptops. From N equals 80, OK, is going to be 80 choose 4. And that's equal to 80 factorial divided by 4 factorial multiplied by 76 factorial. Now when we simplify here, that's going to be the product of 80, 79, 78, and 77. Divided by the product of 432, and 1 are 4 factorial. And that is going to be equal to 1,581,580. In other words, these are the total number of ways to select 4 laptops from 80. Next, we need to figure out the number of favorable laptops, that is selecting only good laptops. Now, the number of favorable outcomes occurs when all 4 selected laptops are good. I know when all of them are good, when all 4 selected laptops. Are good OK Where we already know that 5 are defective, OK. That from The 75 good laptops, OK. So now we know that we're choosing 4 laptops, so that's R, R would be 4, from 75 good laptops. So in this case N would be 75. Then the number of ways to select 4 good laptops from the 75 good laptops would be 75, choose 4. So that's going to be 75 factorial divided by 4 factorial multiplied by 75 minus 4 or 71 factorial. That's going to be the product of 75, 74, 73, and 72, divided by 4 factorial. And when we simplify here, we get that to be 1,215,450. So these are all the ways to select 4 good laptops from 75 good laptops. No, the probability that none of the four selected laptops is defective will be equal to or will be the ratio of favorable outcomes to total outcomes. So the probability then that the shipment is accepted. It is going to be equal to 75 choose 4 or 1,215,450 divided by 80 choose 4 or 1,581,580 and when we solve here we get that to be approximately 0.7685. So there's a 76.85% chance that the shipment will be accepted. If we look back at our answer choices, that means B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

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