"Using a Distribution to Find Probabilities In Exercises 11–26, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.

Typographical Errors A newspaper finds that the mean number of typographical errors per page is four. Find the probability that the number of typographical errors found on any given page is (a) exactly three, (b) at most three, and (c) more than three."

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Step 1: Identify the appropriate probability distribution. Since the problem states that the mean number of typographical errors per page is four, and we are dealing with the number of occurrences of an event in a fixed interval (a page), the Poisson distribution is appropriate. The Poisson distribution is defined by the parameter λ (lambda), which represents the mean number of occurrences. Here, λ = 4.

Step 2: Write the formula for the Poisson probability mass function (PMF). The formula is: P(X = k) = (λ^k * e^(-λ)) / k!, where k is the number of occurrences, λ is the mean, and e is the base of the natural logarithm (approximately 2.718).

Step 3: Solve part (a). To find the probability of exactly three typographical errors (P(X = 3)), substitute k = 3 and λ = 4 into the Poisson PMF formula. This gives: P(X = 3) = (4^3 * e^(-4)) / 3!. Simplify the expression to calculate the probability.

Step 4: Solve part (b). To find the probability of at most three typographical errors (P(X ≤ 3)), sum the probabilities for X = 0, X = 1, X = 2, and X = 3. Use the Poisson PMF formula for each value of k (0 through 3) and add the results: P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3).

Step 5: Solve part (c). To find the probability of more than three typographical errors (P(X > 3)), use the complement rule: P(X > 3) = 1 - P(X ≤ 3). Use the result from part (b) to calculate this probability. Finally, determine whether the events are unusual by comparing the probabilities to a threshold (e.g., 0.05).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Poisson Distribution

The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given a known average rate of occurrence. It is particularly useful for modeling rare events, such as typographical errors in a newspaper, where the mean number of occurrences is known. The formula for the Poisson probability mass function is P(X=k) = (λ^k * e^(-λ)) / k!, where λ is the average rate and k is the number of occurrences.

Probability Calculation

Calculating probabilities involves determining the likelihood of a specific outcome occurring within a defined set of possibilities. In the context of the Poisson distribution, this includes finding the probability of exactly three typographical errors, at most three, and more than three. These calculations can be performed using the Poisson formula or statistical software, which can simplify the process and provide accurate results.

Unusual Events

An event is considered unusual if its probability of occurrence is significantly low, typically defined as less than 5%. In the context of the given problem, after calculating the probabilities for the typographical errors, one must assess whether the results indicate unusual occurrences. This assessment helps in understanding the significance of the findings in relation to the average rate of errors reported.

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