If a publisher of nontechnical books takes gains to ensure...

Name: Problem 81, Chapter 2: Probability with Applications in Engineering, Science, and Technology
Uploaded: 2020-12-22T22:26:22-08:00
Duration: 4 min 54 s
Channel: Robin Corrigan
Description: If a publisher of nontechnical books takes gains to ensure that its books are free of typographical errors, so that the probability of any given page containing at least one such error is .005 and errors are independent from page to page, what is the probability that one of its 400 -page novels will contain exactly one page with errors? At most three pages with errors?

If a publisher of nontechnical books takes gains to ensure that its books are free of typographical errors, so that the probability of any given page containing at least one such error is .005 and errors are independent from page to page, what is the probability that one of its 400 -page novels will contain exactly one page with errors? At most three pages with errors?

Key Concepts

Cumulative Probability

Cumulative probability involves summing the probabilities of multiple outcomes. For instance, finding the probability of having at most three pages with errors requires adding up the probabilities of having zero, one, two, and three error pages. This technique is essential for calculating probabilities over a range of outcomes.

Binomial Distribution

The binomial distribution models the probability of achieving a fixed number of 'successes' (in this case, pages with errors) in a specified number of independent trials (pages), each of which has the same probability of success. This concept allows us to calculate the probability of events such as exactly one error-containing page or at most three error-containing pages in a 400-page novel.

Independent Events

Independent events are those whose occurrence does not affect the probability of one another happening. In this context, assuming each page’s chance of having an error is independent validates the use of the binomial distribution, as the error occurrence on one page does not influence the error occurrence on any other page.

Transcript

00:01 In this exercise, we are told that for a certain publisher, the probability of having at least one error on a given page of the book is 0 .005, and that errors are independent from page to page.

00:16 And we are interested in the number of pages in a 400 -page novel that have errors.

00:23 So let's define a random variable x as the number of pages with errors.

00:37 That's at least one error.

00:39 We are given in the question that the probability that an individual page has errors is 0 .005, and the total number of pages in the book that we were going to look at is 400.

00:58 So these meet the criteria for a binomial distribution.

01:02 So each page is a bernoulli trial, where it can either have, there's only two possibilities.

01:08 It either has errors or it does not.

01:09 The probability of success or the probability of having errors is 0 .005.

01:14 And we're also told that errors on each page are independent from errors on other pages.

01:20 So all the pages are, all the bernoulli trials are independent from each other.

01:25 So we can say that the number of pages with errors in a 400 page book then is a binomial based on 400 trials.

01:40 And probability of success, 0 .005.

01:47 Now this also happens to meet the criteria for approximating a binomial distribution with a poisson distribution.

01:55 Anytime we have a large number of trials, in this case 400, and a small probability of success, in this case 0 .005, then we can approximate this binomial with a poisson distribution with parameter mu equals n times p, which equals two.

02:39 And so now on to answering the questions, we are first asked, what is the probability that a 400 -page book contains exactly one page with errors...

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