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.

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.
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