 Tuesday, October 26, 2010

Estimating Probability of Back to Back Lottery Jackpot Winners Using the Poisson Distribution - Part 3 Part 3: Introduction
In our previous article, we provided an example of how the Poisson Distribution could be used to estimate the probability of multiple jackpot winners (Poisson Distribution Example of Use in Lotteries - Part 2). To carry the application of this statistical model forward, we will calculate the likelihood of there being back to back lottery jackpot winners in both Powerball and UK Lotto. We choose these two games because the frequency of winners in these two games vary immensely.

Poisson Distribution Utilization Review
The Poisson Distribution is a tool used to predict the probability of a discreet event occurring. To use it, there must be a clearly defined observed set of outcomes. Those outcomes are summarized and described as a single average. The distribution of varying events therefore becomes a function of this average.

For example, assume that we wish to define the probability that we will observe 3 automobiles queued at a stop light. The traffic signal changes to red only once an hour. From our previous collection of data, we know that the average length of the queue is 4.8 cars per hour. Substituting these numbers into our Poisson equation, we find that there is a 15.2% chance that the following queue will contain 3  cars.

Now we shall apply these same principles to estimating the probability of a lottery jackpot being won two consecutive drawings in a row.

Example 1: Estimating the Probability of Back to Back UK Lotto Jackpot Winners.
The UK Lotto is the national lottery of the United Kingdom. Since it is a 6/49 game, the approximate number of combinations is about 14 million. By U.S. standards, this is rather small. Being the country's primary game, the average drawing ticket sales range from approximately 14 to 32 million.

Since ticket sales meet or exceed the number of combinations, the UK Lotto jackpot is won on an average of every 1.283 drawings. To calculate the likelihood of there being successive jackpot winners, we must reduce this average by one (to 0.283), and solve for the 0 (zero) event. In effect, we do this to change from a one base to a zero base.

Solving, we find that there is a 75.4% chance that two UK Lotto jackpots will be won in two consecutive drawings. By comparison, we calculated that back to back winners occurred 77.9% of actual time.

Example 2: Estimating the Probability of Back to Back Powerball Jackpot Winners.
By comparison, Powerball is one of two national lotteries of the United States. Its format requires players to correctly pick 5 of 59 white balls and 1 of 39 Power balls in order to win the jackpot. Expanding this out, we find that there are over 195 million possible combinations. Since this is so large, the jackpot is not won as often as the UK Lotto.

Summarizing Powerball drawing results from 2001 to present, we learn that there are approximately 8.95 drawings between jackpot winning draws. Converting this average to a zero base (7.95 average) and solving for the 0 event (back to back winners), we calculate that there is only a 0.04% chance that there the jackpot will be won in two sequential drawings.

By counting the actual number of times this has occurred in Powerball, we find this happened only 7 times since 2001, or 0.68% of the time.

Conclusion
Comparing the expected probabilities derived from the Poisson distribution to the actual number of occurances, we conclude that the statistical results of back to back winners is a fairly good approximation of reality. While the Poisson distribution underestimates reality in both cases, we believe that the results obtained can be confidently used to predict these lottery events.

Tuesday, October 19, 2010

Poisson Distribution Example of Use in Lotteries - Part 2 Part 2: Introduction
Last week we introduced the Poisson Distribution stating that it is used in statistics for quantifying the probabilities of discreet  random events. In our post Using Poisson Distribution to Understand Lottery Events - Part 1, we described its mathematical properties, formula, and variables. In this article, we will provide an example of how the Poisson Distribution can be used to help us understand events related to lotteries.

Example: Estimating the Probability of Multiple Jackpot Winners.
In this example, we will estimate the the probability that there will be 0, 1, 2, ... 5 winning tickets in tonight's Mega Millions lottery drawing which offers an annuity jackpot of \$84 million.

In order to do this, we must first calculate the "expected number of winners" as defined in How to Analyze the Lottery. There, we learn that we need 2 pieces of information:
1. The expected number of ticket sales, and
2. The total number of unique combinations.
Using Mega Millions Lottery Sales By State, we find that last Friday's ticket sales were 25.4 million when the jackpot was \$72M. Using a simple proportion, we will expect tonight's tickets sales to be 29.6 million. Then, from our Lottery Power Picks website, we find that there are 175.7 million combinations. By dividing the expected number of ticket sales by the total number of available combinations, we calculate the "expected number of winners" to be 0.169. In Poisson Distribution terms, this number becomes the known mean, or constant variable r = 0.169

Next we construct a table where: the mean variable r remains constant; and the variable k (which represents the random number of winners) ranges from 0 to 5; and, the associated Poisson probability is solved as variable p(k).

 r k p(k) 0.169 0 0.8445 0.169 1 0.1427 0.169 2 0.0121 0.169 3 0.0007 0.169 4 0.0000 0.169 5 0.0000

Thus reading our table, we learn that there is: an 84.45% chance that there will be no winners in tonight's Mega Millions drawing; a 14.27% chance that there will be one winner; a 1.21% chance that there will be 2 winners; a 0.07% chance that we will have 3 winning tickets; and virtually 0.0% chance that there will be four or more winners.

So, we'll look tomorrow at the Mega Millions drawing results to determine which of our random scenarios occurred.

Tuesday, October 12, 2010

Using Poisson Distribution to Understand Lottery Events - Part 1 Introduction
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 where:
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.

Tuesday, October 5, 2010

Next Weeks Bimonthly Article Announcement - Poisson Distributions

I've been reading a lot about probabilities lately and have discovered a few great articles pertaining to Poisson Distribution.

So, I've started working on a post which will be titled:

Using Poisson Distribution to Understand Lottery Events

I thought this would be easy, but the more I've researched it, the more I am learning. So for now, I expect to have this complete and published next week.

JL................

Monday, October 4, 2010

Euro Millions Jackpot Set to €129 Million this Friday

For the 7th time in its history, the EuroMillions Jackpot is above the €100 million level. However, this is not a record.

During 2006, the jackpot reached €180M twice before being won. These were the only times that the jackpot grew naturally from drawing to drawing. All others times the jackpot was artificially set as a bonus level amount.

Since '06, three €130 million jackpots were up for grabs. One was in September 2007 and shared by 3 winners. The other two were in 2008 and there were no winners of either of these.

The jackpot never reached €100M in 2009.

But this is the second time the jackpot was established at €129 million in 2010.

So, lets cross our fingers and hope that one of us wins the EuroMillions €129 Million jackpot this coming Friday.