Defective Computer Keyboards A shipment of 24 computer keyboards is rejected if 4 are checked fo

step 1 We'll start this problem with describing what we know. We know that there are 24 keyboards in th

Defective Computer Keyboards A shipment of 24 computer...

Key Concepts

Sampling Without Replacement

This concept refers to the process of selecting items from a set where each item, once chosen, is not returned to the population. This affects the probability of subsequent selections because the composition of the population changes with each draw, which is a key consideration in many quality control and inspection problems.

Complement Rule

The complement rule in probability states that the probability of an event occurring is equal to one minus the probability of the event not occurring. This principle is particularly useful in cases where it is easier to calculate the chance of the complementary event, and then subtract that value from one to find the desired probability.

Hypergeometric Distribution

This probability distribution is used when sampling without replacement from a finite population. It is essential for calculating the probability of a certain number of successes (or defective items) in a fixed number of draws from a population that contains a known number of successes, which is applicable in determining rejection probabilities in quality control scenarios.

Combinatorial Analysis

This area of mathematics involves counting methods such as permutations and combinations. In probability problems where items are selected from a set, combinatorial analysis is used to determine the total number of ways to choose a sample and the number of favorable outcomes, forming the basis for computing probabilities in discrete scenarios.

Transcript

00:01 We'll start this problem with describing what we know.

00:04 We know that there are 24 keyboards in the shipment.

00:15 We know that six of them are defective.

00:23 So that means there must be 18 of them that are not defective.

00:35 We know that we're going to inspect four of them.

00:41 So if we're inspecting four of them, then when it comes time to construct our probability, distribution, we might get zero that are defective, or one or two, or three, or four.

01:01 And since those are the only outcomes, or they are all the outcomes that potentially can come out of this inspection of four, we know that the probability column will add up to one.

01:16 Now, we're going to send the shipment back if at least one of the four we have, inspect is defective.

01:27 So at least one translates into it could be one, it could be two, it could be three, or it could be four.

01:42 Now when we inspect these keyboards, we're not going to replace them.

01:46 Once we look at it and decide if it's good or bad, we're not going to replace.

01:51 So since sampling is being done without replacement, then we're referring to a hypergeometric distribution.

02:23 And because it's a hypergeometric distribution, we have a formula that we can work with.

02:31 And the formula is the combination of a items taken x at a time, times the combination of b items, taken n minus x at a time, all over a plus b items, taken, n at a time.

03:00 So a in this case represents our favorable, and right now our favorable is finding the defective items.

03:09 So in this case, a is going to be our 6.

03:17 And b in this formula represents our unfavorable, and in this process, our b is going to be the 18 that are not defective.

03:28 So therefore, we have all the components we need to apply this formula.

03:33 We have an a value...

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