An online magazine finds that the mean number of typographical errors per page is five. Find the probability that the number of typographical errors found on any given page is (a) exactly five, (b) less than five, and (c) exactly zero.

Verified step by step guidance

1. Step 1: Recognize that the problem involves a Poisson distribution. The Poisson distribution is used to model the number of events (in this case, typographical errors) occurring in a fixed interval (a page) when the events occur independently and at a constant average rate. The mean number of errors per page (λ) is given as 5.
2. Step 2: Recall the probability mass function (PMF) of the Poisson distribution: P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the mean number of events, k is the number of events we are interested in, and e is the base of the natural logarithm (approximately 2.718).
3. Step 3: For part (a), substitute k = 5 and λ = 5 into the PMF formula to find the probability of exactly 5 errors: P(X = 5) = (5^5 * e^(-5)) / 5!. Simplify the expression to calculate the probability.
4. Step 4: For part (b), calculate the probability of less than 5 errors, which means summing the probabilities for k = 0, 1, 2, 3, and 4. Use the PMF formula for each value of k and sum the results: P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4).
5. Step 5: For part (c), substitute k = 0 and λ = 5 into the PMF formula to find the probability of exactly 0 errors: P(X = 0) = (5^0 * e^(-5)) / 0!. Simplify the expression to calculate the probability.

Key Concepts

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 the number of rare events, such as typographical errors on a page, where the events are independent of each other.

Mean (λ) in Poisson Distribution

In the context of the Poisson distribution, the mean (denoted as λ) represents the average number of occurrences of the event in the specified interval. In this case, the mean number of typographical errors per page is five, which serves as the parameter for calculating probabilities related to the number of errors on any given page.

Probability Calculation

To find the probability of a specific number of events occurring in a Poisson distribution, the formula P(X=k) = (e^(-λ) * λ^k) / k! is used, where P(X=k) is the probability of k events, e is Euler's number, and k! is the factorial of k. This formula allows us to calculate the probabilities for exactly five, less than five, and zero typographical errors on a page.