Poisson Distribution w/ book errors

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The discussion revolves around applying the Poisson distribution to determine the probability of typing errors in a manuscript, where 14% of the pages are error-free. Participants clarify that in this context, 'p' represents the probability of a single error, while 'n' is not necessary for the calculations at this stage. The key formula for the probability of no errors is discussed, emphasizing the relationship between the rate of errors (λ) and the probability of observing zero errors. The conversation highlights the need to focus on the given probability of zero errors to derive further probabilities for one or more errors. Understanding these concepts is crucial for solving the problem accurately.
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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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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?
 
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.
 
What's the probability of no errors on a page, keeping in mind you've been told that the manuscript is lengthy?
 
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?
 
well 14% of pages have 0 errors... 14%
 
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?
 

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