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How to Calculate Probability Using the Poisson Distribution

By July 20, 2022No Comments

Calculating Probability Using the Poisson

Students who graduate from the lean Six Sigma Green Belt program are exceptionally good at analyzing, and presenting sets of data. One of the key formulas that students will need to know is how to calculate the probability using poisson distribution. This distribution is specifically tailored for the world of probability. Discrete data is characterized and only non-conformities are counted.

This formula is specifically created to identify defects in data sets. As a project manager, this is an extremely important part of the quality assurance process. In this article, we will explore how the Poisson distribution works in the world of project management.

The Background

This distribution comes with a quite impressive historical background. This distribution was established by a mathematician named Simon Dennis Poisson. This distribution became essential when identifying mistakes and challenges that occur in data sets.

When Should I Use The Poisson Distribution?

The Poisson distribution is used to model the number of times a mistake has occurred in a particular data set. For example, if you are a developer, the number of failed app downloads would count towards this distribution. This distribution can be used to determine the likeliness of occurrences per unit. Back to our original example, you could use this distribution to identify challenges per user, per download, or per cell phone carrier. This method is especially useful for analyzing data when there are few trials and the likelihood of failure is high.

Understanding The Poisson Experiment

An experiment containing this methodology will have the following characteristics:

  • The trial produces outcomes that can be considered failures or successes
  • The typical success rate of these occurrences
  • The likelihood of success is appropriate when compared to the region size
  • The likelihood of success is less the smaller the region is

The Poisson Formula

Here is the formula for the Poisson Distribution:

f(x) = P(X=x) = (e-λ λx )/x!

Where:

P(X=x) = Probability of exactly (x) occurrences in an interval

Λ (lambda) = Mean number of occurrences during interval

X = number of occurrences desired

e = base of the natural logarithm

Understanding the elements of this formula is essential before using it to analyze your data sets. It is important to review all of the different symbols prior to starting your trials.

We recommend downloading this distribution table online, to have beside you as you begin analyzing your data.

Visualizing The Poisson Distribution

In order to get to the root of a problem, danger, or challenge is identified in a dataset, you will need to examine the records. The issue typically lies in the number of problems occurring in a specific trial. In this example, we will assume that the average number of problems occurring each week is 5. We will use this data to now calculate the likelihood of:

  • Having more than three accidents occur per week
  • Having less than two accidents occur per week

What Happens When There Are Fewer Than 2 Accidents Per Week Using the Poisson Distribution?

First and foremost, we will need to calculate the probability of having less than two accidents and a specific week. The likelihood of less than two signifies the possibility of no accidents and the second possibility of 1 accident occurring in the week’s time. Using everything you know about the rules of probability, it is now time to customize the formula to set this particular problem.

Your main variable should refer to the number of occurrences that are wanted out of the trial. For example, you can use P (x equals 0) + P (x equals 1). The main components of the formula signify the different factorials of 0 + 1. Keep in mind to always use an exclamation mark when using a 1.

Calculating the Probability of More Than Three Accidents

Next, we will move forward with calculating the likelihood of more than three accidents occurring in a single week. In this model, we will use the distribution P (x is greater than 3). The likelihood of more than 3 indicates the first probability, the 2nd probability of one incident, the third probability of 2, and the fourth probability of 3 incidents. Due to the rules of probability, you must abide by the rules of customization.

Revisiting The Formula

The X variable will correspond to the number of occurrences expected. The first equation will appear as ‘1 – {P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)}’. The focal point component is repeated four times in four separate segments. The first segment is ‘P(X=0)’. The second segment is ‘P(X=1)’. The third segment is ‘P(X=2) and the fourth segment is ‘P(X=3)’.

Be sure to refresh yourself on the rules of factorial notation prior to understanding this equation. It is crucial for you to be able to communicate this data to your team and project stakeholders!

Don’t let the formulas and jumble of numbers scare you! This equation may seem difficult, but with practice, you will be a pro!

Keep in mind that there are a few different software tools out there that can help perform the distribution calculations on your behalf. That said, it is still important for you to understand the basic functionalities of the equations, so that you can check the work of the tools. We hope this article helped you better understand how to calculate probability using this specific distribution!

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