Poisson Pit: Poisson Probability Calculator

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. 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 idiotic judiciary will conclude that “there must be something going on” with these buildings. And they are right, there is – it is called “Chance”.

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

To continue: Age, ethnicity, diet, and many other things affect the likelihood of developing breast cancer. Makes no difference. Even when these are accounted for, chance predicts these "surprising" results, and human nature predicts that they will be confounded with causality.

For another example, 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.)

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.