## Who’s Going Through a Toll Booth First? Ask Poisson

**Introduction (via Zachary Turpin)**

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.

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