機率 Poisson random 問題

Question

A particular typist makes an average of 0.3 typos per page.

Let X = the number of pages the proof-reader checks until

he finds the 5th pages with at least 3 typos.

這種要怎麼列出式子 我碰到這種後面說條件的我就看不太懂了

Answer 1

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First find the probability that a page has at least 3 typos:

= 1 - (probability of 0 typo + probability of 1 typo + probability of 2 typos)

= 1 - [ 0.3^0/0! + 0.3^1/1! + 0.3^2/2! ] / e^(0.3)

Let the answer be P, then the expected number of pages until you find one such page is 1/P. The expected number of pages until you find 5 such pages will be 5/P.

For further info, X follows a negative binomial distribution.

The probability that the proof-reader finds the 5th page with 3 typo on the Nth page is:

P^5 * (1-P)^(N-5) * (N-1)C(4).