A bookstore owner examines 5 books from each lot of 25 to...

Name: Problem 18, Chapter 5: Elementary Statistics a Step by Step Approach
Uploaded: 2020-07-10T20:36:11-08:00
Duration: 8 min 21 s
Channel: Evelyn Cunningham
Description: A bookstore owner examines 5 books from each lot of 25 to check for missing pages. If he finds at least 2 books with missing pages, the entire lot is returned. If, indeed, there are 5 books with missing pages, find the probability that the lot will be returned.

A bookstore owner examines 5 books from each lot of 25 to check for missing pages. If he finds at least 2 books with missing pages, the entire lot is returned. If, indeed, there are 5 books with missing pages, find the probability that the lot will be returned.

Key Concepts

Hypergeometric Distribution

This distribution models the probability of a specified number of successes in draws from a finite population without replacement. It is particularly useful when the total number of items, the number of successes in the population, and the sample size are all fixed, making it ideal for quality control scenarios where items are inspected from a batch.

Combinatorial Analysis

This concept involves counting the number of ways in which an event can occur using combinations. In probability problems like this, combinatorial analysis is used to determine the number of favorable outcomes as well as the total number of outcomes by calculating binomial coefficients, which are central in computing hypergeometric probabilities.

Sampling Without Replacement

This concept refers to drawing items from a population where each item, once drawn, is not returned to the pool for subsequent draws. This affects the probability of subsequent selections, as the population size decreases with each draw. It is a key aspect in determining the success probability in quality control and acceptance sampling scenarios.

Acceptance Sampling

Acceptance sampling is a quality control method used to determine whether to accept or reject a group of items (a lot) based on inspecting a sample from it. Decisions are made based on criteria such as the number of defective items found in the sample. This method helps in efficiently managing quality by reducing the need to inspect every item in a batch.

Transcript

00:01 Let's start this problem by writing down what we know.

00:05 We know that there are indeed five books with missing pages.

00:20 We know that in a lot of books, there's a total of 25 books.

00:32 So if there are five that have some pages missing, that must mean that there are 20 books without missing pages.

00:51 And in this particular problem, we are going to select five books.

00:56 We're going to look for missing pages.

00:59 And if he finds at least two books with missing pages, then the entire shipment of books is going to be returned.

01:08 So since we are taking a sample size of five books, our n is going to be five.

01:15 Now, when we take those five books, we're going to take a book out of the supply.

01:19 We're going to inspect it.

01:21 And if we find pages that are missing or not, we're not going to put that book back into the, pile before we select our next one.

01:30 So when we do sampling without replacement, we are dealing with hypergeometric distributions.

01:50 And again, it's when we do not replace.

02:04 And if that's the case, we have a formula that we're going to follow.

02:08 And the formula says p of x equals.

02:14 We're going to combination, a, items, taken x at a time.

02:22 We're going to multiply that by b items taken n minus x at a time, and we're going to divide by a plus b items taken n at a time.

02:37 So a is going to be referencing the books with missing pages, and b is going to represent our books without the missing pages.

02:53 We already defined what our n was.

02:55 Our n is going to be five.

02:57 Now, in this problem, we are looking for at least two books.

03:02 So if i translate what at least two books means, means that x could be two or three, or four, or five.

03:22 So let's think of what, about what the probability distribution would look like...

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