In a lengthy manuscript, it is discovered that only 13.5...

In a lengthy manuscript, it is discovered that only $13.5$ percent 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 error.

Key Concepts

Poisson Distribution

The Poisson distribution is a discrete probability distribution that models the number of times an event occurs within a fixed interval of time or space. It is particularly useful when the events are independent and occur at a constant average rate, making it a common choice for modeling rare events such as errors on pages.

Probability Mass Function (PMF)

The PMF of a Poisson distribution gives the probability of observing exactly k events and is formulated as P(k) = (?^k * e^(??)) / k!, where ? is the average rate of occurrence. This function is central for calculating the likelihood of specific counts of events, providing a basis for further analysis when dealing with count data.

Parameter Estimation Using Observed Probability

When given an observed probability, such as the proportion of events with zero occurrences, the parameter ? can be estimated by solving the equation for k = 0, namely e^(??) = observed probability. This estimation process is a key step in applying the Poisson model to real-world data, allowing for the prediction of probabilities for other outcomes.

Application of the Poisson Model to Specific Outcomes

Once the average rate of occurrence (?) is determined, the Poisson model can be used to compute the probability for any specific number of events. By substituting the desired count into the PMF, one can, for example, calculate the probability of having exactly one error, demonstrating the practical use of the model in predictive scenarios.

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