Variously called the Poisson Distribution, Poisson Law of Large Numbers, or Poisson Distribution for Rare Events, the probability function for this discrete distribution of one parameter is given above. The Poisson distribution is usually employed for modeling systems where the probability of an event occurring is very low, but the number of opportunities for such occurrence is very high. Take radioactive decay, for example. Or rare disease clusters.

Example 1. Assume that on average, each square mile of the earth's surface is struck by one meteor of a certain size each year. How many square miles of the earth's surface will be hit by no meteors at all, and how many will be hit by more than 5? Click on 1 in the "m (expected occurrences per trial)" box and on 200,000,000 [square miles of the earth's surface] in the "Number of Trials". Then click the COMPUTE button. The answer is that 73,575,888 square miles will escape bombardment totally, while 731,969 square miles will be hit five or more times. (200,000,000 - 199,268,031, the number of square miles hit 4 or fewer times.)

Example 2. The celebrated Motel Carswell forfeiture case gives another example. In that case, U.S. District Attorney Carmen Ortiz brought a civil forfeiture action against a motel that had 15 drug-related incidents in a 14-year period. Given that there are ~50,000 hotels and motels in the United States, assuming a 15-year interval and an average of one incident per hotel every three years (M=5), we would expect, with 50,000 hotels and motels, 24 would have 14 incidents, 8 would have 15, 2 would have 16, and 1 would have 17. A federal magistrate wisely dismissed the case, but only because a nearby Walmart had a similar number of drug offenses occur there --  not because the magistrate had any grasp whatsoever of probability theory or statistics, indeed the contrary.  What a bunch of idiots.  This case illustrates the pervasive innumeracy of the U.S. justice system -- a U.S. district attorney and magistrate stupid enough to think that a departure from group averages indicates causality.

m
(Expected
occurrences
per trial):

 
Number
of
trials:

x
Number of
Occurrences
p(x;m)
Probability of a trial having this number of occurrences
Cumulative probability of a trial having this number of occurrences or less Expected trials with this number of occurrences Cumulative number of trials with this number of occurrences or less

Example 3. Assume 10,000 office buildings, each building having 100 female employees, and each female employee having a 1/100 chance of developing breast cancer in a given year. How many buildings will have no cases at all in a given year? And how many will have six cases? Click on 1 in the "m (expected occurrences per trial)" box and on 10,000 in the "Number of Trials" box. Then click the COMPUTE button. The answer is that 3,679 building will have no cases at all, and five buildings will have six cases. Human nature being what it is, people will conclude that there must be something wrong with these five buildings, and seize on some unusual characteristic -- proximity to a power line, presence of mold, or some other such nonsense -- as an explanation. Lawsuits will be filed, and our slightly demented judiciary will conclude that "there must be something going on" with these buildings. And they are right, there is something going on -- it is called "Chance".

Of course, age, ethnicity, diet, and many other things affect the likelihood of developing breast cancer. Doesn't matter. These variables increase the odds of breast cancer in any given individual relative to group averages, but they are not fully deterministic in any given time interval. A large element of chance remains. Thus, if we had 10,000 buildings with 100 women in each and all women had the BRCA1 gene, we would expect more cases of breast cancer overall, but with similar extremes in individual buildings. Likewise, in the Motel Caswell case, we might expect indexpensive motels in high-crime areas to have more crimes than expensive motels in low-crime areas, but ceteris paribus, we would expect similar extremes in individual motels. Chance predicts these "surprising" results, and human nature, along with the U.S. failure to teach probability and statistics in our schools, predicts that they will be confounded with causality.

Finally, please note that when a 1 or zero is shown in the cumulative probability or expected trials field, it does not mean exactly zero or 1, just that the number is closer to zero or 1 than any number that can be displayed in the space available without resorting to floating point notation.




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