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When statisticians calculate the likelihood of a random event, they sometimes use something called a “Poisson distribution.” Web-search this term. For the less-than-mathematical, the explanations that come up may read like migraine-inducing gibberish. Which is unfortunate, really, since the PD is so useful that it deserves a simple explanation.

A Poisson distribution is, at its root, a mathematical equation that uses a little data, supplied by you, to determine the odds a random event will occur X times in a given parameter (in an area, maybe, or a span of time). The event can be almost anything, as long as A) it is random, B) you know how often it occurs on average, and C) it occurs independently of the time since the last event. These event-strings, called “Poisson processes,” include things like:

• the number of raindrops to fall on a square of sidewalk
• the odds of getting a bulls-eye in darts
• the number of calls to pass through a 911 center in an hour
• the number of typos made per page
• the odds of winning at roulette

A characteristic of the Poisson process is the ease with which its mechanism may be understood, and the difficulty with which it is predicted. Traffic, for example: the distribution of cars on the road is easy to conceptualize. When a lot of cars hit the road at once, there’s traffic. It is not, however, easy to predict if the traffic on a Wednesday will become a traffic jam or how long a jam might last. With collectible numbers like average cars on a highway at 5:00 and average number of accidents per day—and a Poisson calculation—the likelihood of those events become clearer. Other Poisson processes are similarly simple and chaotic—how many people will be in a given grocery line when you arrive? How many phone calls will you receive in a day? Poisson distributions sidestep a bit of the chaos relying solely on average rate of occurrence.

Siméon-Denis Poisson, a French mathematician, developed the distribution in 1830. Its purpose, initially, was to determine a gambler’s odds of winning a very unlikely game of chance (say, roulette) if he or she played it enough times. Since then, the PD has been used to predict (or model) any number of random phenomena. Perhaps its most famous (20th-century) use was in determining the accuracy of Hitler’s V-2 rockets.

During WWII, the worst of Nazi Germany’s rocket-propelled bombs, the V-2, would (when fired from Holland) climb over 50 miles into the atmosphere, shut off, fall, and explode somewhere in London. The V-2 was an instrument of fear: its death toll was low, but fear of the V-2 (which traveled far faster than the speed of sound, and came to Earth with no warning) was rampant. In 1946, a British statistician named R.D. Clarke analyzed the pattern of V-2 explosions in London. He wanted to know if they landed in targeted areas or randomly. Clarke theorized that, if V-2’s fell randomly, their landing pattern would resemble a predicted Poisson distribution. By comparing a map of London (divided into little squares) to a distribution based on average-explosions-per-square, he determined that V-2’s were not guided missiles after all. They fell randomly.

Here is Poisson’s equation: P(X) = (nˆX)(eˆ−n)/X!

And a breakdown:

P(X)—the probability an event might happen X times (the desired odds, that is)

X—the number of times the event might happen (e.g. if you want to know the odds of an event happening 3 times in a given span, X=3)
n—the average number of times an event happens (the only other information needed)
e—the base of natural logarithms, or about 2.71828
!—a factorial, or multiplying a number by all the natural numbers below it: for example, 3! (or “three factorial”) = 3x2x1 = 6, and 5! = 5x4x3x2x1 = 120

The key to using a PD is knowing n, the average frequency of the event. And as long as an average is based on extensive study—say, 1,000 hours spent recording roulette outcomes—then its PD will yield viable probabilities. If an average is really just a guess—e.g. assuming that a summer shower drops, oh, about 5,000 raindrops per sidewalk square—plugging it in will only yield a guess-based-on-a-guess. With a hard average, though, a PD can be remarkably predictive.

An example. Say a typist makes an average of 3 typos per page. Based on that average, what are his or her predicted odds of making 1 typo on a given page? What about 6 typos?

The equations, with the average (n) and desired number of times (X) plugged in, look like this:

P(1) = 31(2.71828-3)/1! = 3(0.04979)/1 = 0.14937 = 14.9%
P(6) = 36(2.71828-3)/6! = 729(0.04979)/720 = 0.05041 = 5.0%

If you take the results of all the possible typos from 0 to 6, you have a “distribution,” or graph, that resembles a warped bell curve. In this case, the likeliest outcomes for a random page are 2 or 3 typos—both have a 22.4% probability. On either side the odds trail off, and the more expected typos, the lower the odds. For someone who makes an average of 3 typos per page, the predicted odds of making 20—or P(20)—come out to only 0.00000000007%.

“Predicted odds” is an important phrase. Many Odds Statements at Book of Odds are historical: made based on how often something has happened in the past. In the case of typos, BOO might track how many are made on each and every page, then create a table containing data on every existing typos-per-page number. So, out of 10,000 pages of typed material, say our typist makes 3 typos on 2,500 of those pages. 2,500 triple-typoed pages / 10,000 total pages = 1/4. The resulting BOO Odds Statement would appear as “the odds Our Imaginary Typist will make 3 typos on a given page are 1 in 4.”

A Poisson distribution works differently. Rather than keeping tabs on each outcome’s numbers and then constructing fractions, a PD merely requires an average. So, one would merely add up all the typos—say, 30,000—and divide by total pages (10,000, or “a myriad”) to get an average: 3 typos per page. This average is then plugged into the PD, as above.

So while the example above is illustrative, it isn’t typical. PD’s are more often used to predict the odds of events that are virtually uncountable on a large scale, or at least unfeasible to count. Typos are relatively easy to keep track of; a computer program could tally them. The number of raindrops landing on a million sidewalk squares, however, or a person’s number of lottery wins if they were to play 3 million times...

PD’s are useful, in other words, for predicting a large number of possible outcomes, each of which is rare. A single raindrop’s odds of hitting a single spot on the sidewalk are astronomical. But the odds of a given number of drops landing on a random square meter of sidewalk—that’s a job for a PD, provided one knows, or can find, the average. Whether it’s working out radioactive decay rates or calculating the distribution of galaxies in the Universe, the Poisson distribution is another useful tool in a statistician’s belt.