A first draft of a 400 -page manuscript has 36 typos that...

A first draft of a 400 -page manuscript has 36 typos that are spread randomly across the pages. Assume the number of typos per page follows the Poisson distribution. a. What is the probability that there is one typo in the first 10 pages? b. What is the probability that there is one typo in the first 20 pages? c. What is the probability that there are four typos in the first 50 pages? d. What is the probability that there are 12 typos in the first 110 pages?

A first draft of a 400 -page manuscript has 36 typos that are spread randomly across the pages. Assume the number of typos per page follows the Poisson distribution.
a. What is the probability that there is one typo in the first 10 pages?
b. What is the probability that there is one typo in the first 20 pages?
c. What is the probability that there are four typos in the first 50 pages?
d. What is the probability that there are 12 typos in the first 110 pages?

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Key Concepts

Rate Parameter (?) and Scaling

The rate parameter, denoted as ?, represents the expected number of events per unit interval (in this case, per page). When extending to a larger interval (multiple pages), the overall expected number of events is obtained by scaling the per-unit rate by the number of units (pages). For example, if the average typos per page is ?, then for n pages the expected rate becomes n?.

Probability Mass Function

The probability mass function (PMF) of the Poisson distribution is used to compute the probability of observing exactly k events in an interval. It is given by the formula: P(X = k) = (?^k * e^(??)) / k!, where ? is the expected number of events in that interval. This formula is essential for calculating the probabilities of a specific number of typos occurring in selected intervals of pages.

Poisson Distribution

The Poisson distribution is a discrete probability distribution that describes the number of events occurring within a fixed interval of time or space, under the assumption that these events occur with a constant average rate and independently of each other. It is commonly used to model random occurrences such as typos across pages in a manuscript.

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