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.

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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.