The Poisson distribution is usually employed for modeling systems where the probability of an event occurring is low, but the number of opportunities for such occurrence is high. Reason.

Example: During the V-1 attacks on London, rumors spread that certain points were being precisely targeted. Given the poor accuracy of the V-1, this seemed unlikely, but to test the assertion, R. D. Clarke divided an area of South London into 576 sections of ¼ square kilometer each. The total number of hits in the area was 537 -- 0.932291667 per section. Clarke then used the Poisson distribution to calculate how many sections could expect 0, 1, 2, 3, 4, or 5 or more hits, if the V-1's were striking at random throughout the area. The numbers very closely matched the actual distribution, suggesting that the observed hit clusters resulted by chance. Click here to replicate Clarke's results using this calculator. Click here to see Clarke's article.
m
Expected
occurrences
per trial:
n
Number
of
trials:
*****Results will be displayed here when you click the COMPUTE button*****


Example: 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 for the 15-year period), we would expect, with 50,000 hotels and motels ("trials"), 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. A better reason might have been to reject the prosecution's argument as just another airhead example of the "Law of Averages" fallacy.

Example: Assume 10,000 office buildings, each building having 100 workers, and each worker having a 1/100 chance of developing nostril spasms in a given year. How many buildings will have no cases at all in a given year? And how many will have six cases? Enter 1 in the "m (expected occurrences per trial)" box and 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.

Caveat arithemeticus: Please note that numbers displayed in the result table show 10 significant digits and scale from about 10-322 to 10322. Columns 2 and 4 [p(x;m) and expected trials with p(x;m)] approach but never reach zero, just showing successively smaller exponents. Cumulative probability approaches 1, and cumulative trials approaches N; once the significand for these rounds to 1 or N, they just show 1 or N. Display ends when one or more factors go out of range, or after 1,000 rows are displayed, whichever occurs first. The name and JavaScript value of the out-of-range factor is shown. A JavaScript value of "infinity" does not really mean infinity, just a positive number beyond the range of JavaScript's floating point representation. Also, please note this calculator can be invoked with a querystring, specifying m and n, for example http://www.anesi.com/poisson.htm?n=576&m=.932291667.
Notes

V-1 Example. The same distribution could have been produced if the Germans had very precise targeting capabilities and intentionally threw extra V-1's at marginal sectors to give the impression of a random distribution. That was not the case, but it is worth observing that devious adversaries do things like that (to conceal the true accuracy potential of a weapon, for example).

Motel Carswell Case. Is it reasonable to expect that motels in high-crime areas will have more drug deals than motels in low-crime areas? Possibly. Or perhaps enhanced police surveillance in high-crime areas might cause drug dealers to move their activities to motels in low-crime areas where surveillance is relaxed. It's a complex question. Typically this question would be addressed with a multivariate model, but no doubt a half-buttocked, misspecified model with deficient data, omitted variable bias, ecological fallacies, and a ton of heroic assumptions, bizarrely passed off as establishing a causal relationship.

The real point is that in a case like this, the first question we should ask is, "What reason do you have to believe that this departure from population means is the result of nefarious behavior, and not just chance? If you flip a coin ten times, are you really stupid enough to be amazed if you don't get 5 heads?"

Nostril Spasms. Same observations as for the Motel Carswell Case. The occurrence of many nostril spasms in one building might result from some causative agent linked to the building, but assuming that many nostril spasms in one building imply a causative agent linked to the building is nonsense.


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