Poisson Distribution w/ book errors

In summary, the problem involves a lengthy manuscript with a 14% probability of a page having no typing errors. The number of errors per page is assumed to follow a Poisson distribution, with the mean and variance being determined by the probability of a single error and the number of opportunities for that error to occur. The problem asks to find the percentage of pages with exactly one error, at most two errors, and two or more errors, as well as the mean and variance of the number of errors per page. The formula for a Poisson distribution is used, with λ representing the product of the probability of a single error and the number of opportunities for that error to occur. The concept of a Poisson process is also introduced, with a page
  • #1
joemama69
399
0

Homework Statement



In a lengthy manuscript, it is discovered that only 14% of the pages contain no typing errors. If we assume that the number of errors per page is a random variable with a Poisson distribution, find the percentage of pages that have: Exactly one typing error, At the most 2 typing errors, Two or more typing errors. Also compute the mean and variance of the number of typing errors per page.


Homework Equations





The Attempt at a Solution



I know for a Poisson distribution np=λ

the problem states that p=14% of pages with 0 errors, but don't I also need to know 'n' which would be the number of pages? Anyone got a hint?
 
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  • #2
joemama69 said:
the problem states that p=14% of pages with 0 errors,
No, that's not what p is here. In a Poisson distribution, p is the (very small) probability of a single error and n is the (very large) number of opportunities for the error to occur. There can be hundreds of errors on a page.
Suppose there are N such opportunities per page, each occurring with prob p, independently, and λ = pN. What is the probability of exactly k errors on a page?
 
  • #3
Im not getting it... It just seems like there's not enough information. Don't we need to know the probability of an error and the number pages to find np. I must be missing something.
 
  • #4
What's the probability of no errors on a page, keeping in mind you've been told that the manuscript is lengthy?
 
  • #5
joemama69 said:
Im not getting it... It just seems like there's not enough information. Don't we need to know the probability of an error and the number pages to find np. I must be missing something.

Don't worry about the number of pages for the moment.
Compare this to the more usual setting for a Poisson process, something that happens over a continuum, like time. Think of a page as a period of time, T, and the errors as events that occur randomly in time at a rate λ. What is the probability that no events occur in time T? What value are you given for that probability?
 
  • #6
well 14% of pages have 0 errors... 14%
 
  • #7
joemama69 said:
well 14% of pages have 0 errors... 14%

Right, but what formula can you write using T and λ for the same thing? I.e. what is the probability of no events in time T?
 

FAQ: Poisson Distribution w/ book errors

What is Poisson Distribution and how does it relate to book errors?

Poisson Distribution is a mathematical concept used to model the probability of a certain number of events occurring within a specific time or space. In the context of book errors, it can be used to determine the likelihood of a certain number of errors appearing in a given number of books.

How is the Poisson Distribution equation calculated?

The Poisson Distribution equation is calculated by taking the mean or average number of events (λ) and plugging it into the following formula: P(x) = (e^-λ * λ^x) / x!, where x is the number of events and e is Euler's number, approximately equal to 2.71828.

What is the significance of the mean (λ) in Poisson Distribution?

The mean (λ) in Poisson Distribution represents the average or expected number of events within a certain time or space. It is a crucial parameter in calculating the probability of a specific number of events occurring.

Can the Poisson Distribution accurately predict the number of book errors?

The accuracy of Poisson Distribution in predicting the number of book errors depends on the assumption that the occurrence of errors is random and independent of each other. If this assumption holds true, then the distribution can provide a good estimate of the probability of a certain number of errors appearing in a given number of books.

How can Poisson Distribution be used to improve book production and quality control?

By understanding the probability of book errors through Poisson Distribution, publishers can implement quality control measures and production strategies to reduce the likelihood of errors appearing in their books. This can lead to improved overall book quality and customer satisfaction.

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