If you were a horse soldier in the Prussian Army in the late 1800s, you were obviously not at any risk of dying in an automobile accident, but there was a persistent, yet low-risk, problem with soldiers dying from being kicked by their horses. Polish/Russian statistician Ladislaus Bortkiewicz famously found that these “random” deaths did indeed form a pattern, in this case one that had been theoretically described earlier in that century by French mathematician Siméon Denis Poisson,  whom we met in an earlier post about his Law of Large Numbers, and how it relates to infrequent, and even “lottery level” probability events.
When you add time into the probability equation, which happens in nature all the (ahem) time, this special pattern of randomness, now called the “Poisson distribution,” begins to show up everywhere. Think for a moment about how your skin cells replicate. You don’t see it happen, but you know there is always new skin replacing the dead cells that are being washed or rubbed off your body daily.
This cell replication process is called mitosis. Some cells, like muscle fiber, almost never replicate, while a patch of living skin cells under a microscope can replicate at a rate that you can observe happening. The pace of replication will vary depending on the size of your patch. Nothing will appear to be happen, and then….
Wait for it … wait for it …
Suddenly you see a cell divide into two cells. Then it is back to waiting again, which may be a very short period of time, or even instantaneously, or it may be a longer time. But if you were to chart your count of replications every minute, you would likely come up over time with a graph that looks something like this:
Now this looks somewhat like a classic “normal” bell-shaped curve, but squashed against one end. In this case, our patch of skin is averaging about two replications per minute, but an exact count of two happens only about 27% of the time. About 13% of the time, nothing at all happens during that minute, but about one in 100 times, we might see six replications within one minute. Seven or eight replications are rare, but still possible counts. 
These natural processes, where events happen randomly over time are called “Poisson processes,” and we see them outside of biology as well. We could construct a similar graph to count the telephone calls coming into a help desk and we would most likely see Poisson distribution forming over any small interval of time. In this case, the calls might come at, say, an average of two per minute, but a significant number of time the phone does not ring at all during that minute. And sometimes we get five or six calls coming in seemingly at the same time.
I would say that you can blame Simeon Poisson the next time no one answers your hotline support call, but any customer service manager worth his or her salt knows this distribution well, and has more likely just not staffed enough to handle the inevitability of your call.
Restaurant arrivals often fit this pattern as well, with a bit of a twist that people are more likely to come in pairs or larger groups. But adjusting for that, the same pattern emerges, rising quickly from zero (instantaneously) to the observed average of “hits” in one measured period of time, and then tapering off “asymptotically” toward zero probability. In nature, this distribution can be used to predict radioactive decay or the number of people infected by a flu virus within a particular span of time. You can even use it to predict how many ambulances you are going to need to meet a particular community’s peak demands. We’ll look at that one in more detail in an upcoming post. Stay tuned.
But at its heart, this probability distribution predicts the pace of the propagation of life on this planet, as it has been doing for some 3.5 billion years, and you can use it to walk genetic mutations back through that same time period of history as well. Life itself is a “Poisson process.” We live our lives in the passage of time, and “stuff happens over time.” And so, when a “life process” like mitosis goes awry in a living thing, trouble ensues. One way to look at the probability of cancer occurring, discussed in recent posts, is to find out what is making the Poisson probability rate of cell replication to quicken to a destructive pace.
When we are dealing with time, our expectation of events sometimes get out of whack with reality. We put a huge portent on when certain events appear to happen simultaneously, but if the events follow a Poisson distribution in nature (or “God’s world” if that is your preference), this is statistically more likely to happen than you think.
This suggests the question, “Do important (or bad) things really happen in threes?” That answer will need to wait for another post (but spoiler: “probably” not).
The more interesting question to me is, if so many of nature’s “unpredictable” events nicely fit into Poisson curves, what does that say about the theodicy “Big Question” of “Why me?”
Wait for it … wait for it …
- And a reminder that his name is pronounced “pwa-sawn.”
- We can construct a Poisson curve that looks more statistically “normal,” but only if we extend out the amount of time that we are observing, relative to the pace of replication. In this case, if we were to chart the count of replications in one hour of time, the average would rise to 120 replications and the chart would look quite balanced in its bell shape, because the Central Limit Theorem is beginning to take effect. Note also that technically these examples should be represented by a bar graph, as we are counting discrete events. However, showing it as continuous curve better illustrates both its similarity to, and difference from, a normal curve. Wikipedia shows how the Poisson distribution becomes “less normal” the fewer average number of occurrences (k) measured over a given period of time (represented by the Greek letter lambda):