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When discussing Poisson distributions, probability calculation is a key aspect. It involves determining the likelihood of a certain number of events occurring within a fixed interval of time or space. In these scenarios, the average rate of occurrence (mean) is known, and we aim to calculate probabilities for different outcomes.

For example, if we know the average number of orders is 2 per week, and we want to calculate the probability of receiving three or more orders, we look at events happening in a Poisson-distributed manner. The Poisson probability mass function (PMF) helps us find these probabilities. For any number of orders, k, the PMF is given by the formula:

• \( P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!} \)

where \( \lambda \) is the average number of orders, \( e \approx 2.71828 \) is the Euler's number, and k! is the factorial of k. By using this function, we calculate probabilities for specific numbers of orders.

To determine whether a technician is dispatched, we need to understand the probability of receiving three or more orders. This is calculated by finding \( P(X \geq 3) \) in a given week. Since it's often easier to calculate the probability of having exactly 0, 1, or 2 orders, we use their cumulative probability and subtract from 1 to find \( P(X \geq 3) \).

The process involves these steps:

• Calculate \( P(X=0) \), \( P(X=1) \), and \( P(X=2) \) using the Poisson PMF.
• Add these probabilities together to find \( P(X \leq 2) \).
• Subtract \( P(X \leq 2) \) from 1: \( P(X \geq 3) = 1 - P(X \leq 2) \).

For instance, if the probabilities for 0, 1, and 2 orders are calculated to be \( e^{-2} \), \( 2e^{-2} \), and \( 2e^{-2} \) respectively, then \( P(X \leq 2) = 5e^{-2} \). Thus, \( P(X \geq 3) = 1 - 5e^{-2} \), indicating the chance the technician is needed.

The mean number of orders is crucial as it sets the baseline for the Poisson distribution. It is essentially the average rate of incoming orders over a certain period. In our scenario, the mean for a population of 100,000 is 0.25 per week. For a larger population, such as 800,000, the mean scales proportionately.

To find the new mean for a different population size, we use a simple proportional increase based on population:

• Calculate the ratio of the new population to the original population, \( \frac{800,000}{100,000} \).
• Multiply this ratio by the original mean: \( \lambda = 0.25 \times \frac{800,000}{100,000} \).
• This results in a new mean of \( \lambda = 2 \), or 2 orders per week.

Understanding this mean helps predict how often events (orders in this case) are likely to occur in similar scenarios. It also assists in adjusting calculations for other population sizes or time intervals.