Tuesday, October 12, 2010

Using Poisson Distribution to Understand Lottery Events - Part 1

The Poisson Distribution is a statistical model used to project the probability of the occurrence of discreet events. Recently, we have discovered the use of this model in an article, How to Analyze the Lottery, by John Corbett and Charles Geyer. In it, the authors explain how a Cash/Annuity lottery works by evaluating the probability of single and multiple winners.

Based on their work, we have explored the potential use of this model to understand other lottery events.

Thus, to present this information, we are splitting this discussion into 3 parts:
  • Part 1: Definition of the Poisson Distribution
  • Part 2: Examples of Use
  • Part 3: Comparison of Expected Probabilities Verses Actual Events
Today's article Presents Part 1 of our Analysis.

Definition of the Poisson Distribution
The Poisson Distribution  is a statistical model that expresses the probability of a random event occurring in a fixed period of time when:
  • The there is a known average of occurrences
  • It is possible to count the number of times an event has occurred
  • Each occurrence of an event is independent of the previous results
  • Expected events (except the average) must be a whole positive integer
As a formula, the Poisson Distribution is written as:

Poisson Distribution

Poisson Formula

k = the whole integer expected random event event
r = the known mean or average (often represented as lambda)
p(k) = the solved Poisson Distribution probability of event "k"

Note that depending on the text referenced, the variables may be different, and the representation may be slightly different as well (showing the e^r term on the top as e^-r).

The graph below illustrates a sample Poisson Distribution. The vertical y axis shows the probability of an event happening. The horizontal x axis shows variable random occurrences. Note that the probabilities are skewed towards the left where the average occurs. Additionally, the horizontal axis is boundless. Meaning it must never have a discreet limit.

Potential Uses in Lottery Analysis.
When analyzing the lottery, the Poisson Distribution has several applications. For example, we may use it to quantify the probabilities of:
  • Multiple Winners in any Single Drawing, or
  • The the Number of Drawings before a Jackpot is Won

Next Week's Publication - Part 2
As stated above, this will be a three part series. Next week we will illustrate the use of the Poisson Distribution by showing how to estimate the number of winners and the interval between winning jackpot drawings.

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