Solving Poisson Distribution Problems: Probability of Call Receptions in 30 Minutes

In the world of statistical analysis and problem-solving, a common scenario involves understanding and predicting the number of events within a certain period. This article will walk through a specific example involving Poisson distribution, which is particularly useful when dealing with the average number of calls received over a given time interval. We'll solve the problem step-by-step, provide a practical solution using the Breatter App, and illustrate the process with a sample calculation in Excel. Additionally, we'll explore how to approach a similar problem with the correct number of calls.

Understanding Poisson Distribution

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given the average rate of occurrence, λ (lambda).

Key characteristics of a Poisson distribution include:

The events are independent of each other. The probability of an event occurring in a small interval is proportional to the size of the interval. The probability of two events occurring in the same small interval is approximately zero.

Problem Statement and Context

Imagine a call center scenario where the average number of calls received in a 30-minute period is 15. The problem we need to solve is to find the probability that exactly 10 calls will be received between 10:00 and 10:30. This is a typical application of the Poisson distribution.

Solving with the Breatter App

The Breatter App is a user-friendly tool that simplifies the process of solving statistical problems, including Poisson distribution. Here's how you can use it to find the probability:

Open the Breatter App. Navigate to the section that deals with Poisson distribution. Enter the given average (λ) which is 15 (the average number of calls in 30 minutes). Enter the number of calls (k) you are interested in, which is 10. Select the option for the specific value of k to get the probability. The app will provide the probability that exactly 10 calls are received in the 30-minute period, which typically appears in the first answer box.

For detailed step-by-step solutions, the app is recommended. You can find more tools and resources on the app's website or via a web search.

Calculating with Excel

For those who prefer to work with Excel, you can use the POISSON.DIST function to calculate the probability:

POISSON.DIST(10, 15, FALSE)

This function will give you the probability that exactly 10 calls are received in the given time period:

 (1510 * e-15) / 10!

The result of the above formula is approximately 0.04861, or about 4.861%, which represents the probability that exactly 10 calls are received in the 30-minute interval.

Secondary Scenario: Probability of 9 Calls

If the original problem was to find the probability of exactly 9 calls instead of 10, the process would be similar. Simply substitute 9 for 10 in the POISSON.DIST function:

POISSON.DIST(9, 15, FALSE)

This calculation will yield the probability that exactly 9 calls are received in the 30-minute period, which is approximately 0.032407, or about 3.24%.

Conclusion

Understanding and applying Poisson distribution is crucial for solving problems related to call reception, customer service, and other event occurrences in a fixed interval. Utilizing tools like the Breatter App or Excel can significantly simplify the process. By familiarizing yourself with different methods and tools, you can become more proficient in handling such statistical analyses.