Poisson Distribution w/ book errors

[joemama69]

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?

 

[haruspex]

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?

 

[joemama69]

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.

 

[vela]

What's the probability of no errors on a page, keeping in mind you've been told that the manuscript is lengthy?

 

[haruspex]

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?

 

[joemama69]

well 14% of pages have 0 errors... 14%

 

[haruspex]

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?