In an earlier post about Poisson processes, I teased the question, “Do ‘bad things’ really happen in threes, as the common folk wisdom goes?” The short answer is, “No.”
Okay, there is some wiggle room of basis for this myth for certain familiar Poisson processes, where some random timing effects can cause our minds to perceive a natural grouping of events, and sometimes in threes. Or fours. Or more.
The beginning of the problem with the “bad things” story is that your categorization of which events you will put in the “bad things box” is usually very loose. Next, a logical fallacy called confirmation bias steps in, where your mind adds only what it wants into “bad things box” and disregards other events (or the fourth or fifth event) in order to “prove” yourself correct.
So, depending on how you construct your “box” in terms of time and qualifying events, you can perceive bad things happen in any grouping you want, and in any number you want. “Two things” does not seem significant, so three, then.
When it comes to true Poisson processes, however, there is a very real phenomenon, called Poisson clumping, that can play tricks with your mind, and make it seem like there are unnatural groupings of events. If you want to see how this works mathematically, read on.
Recall that Poisson processes are events that happen randomly over time, yet have a consistent “average count” of occurrences within any specified period of time. If that time period is short enough, and if that count of occurrences is small enough, then a unique “Poisson probability” pattern emerges, looking somewhat like a “normal” curve, but squashed toward left side, which represents zero occurrences happening within the specified timeframe.
Let’s say, in this case, you are the only person answering a customer service hotline that, over the long run, averages four calls within each 20-minute period. The probabilities during each 20-minute period look like this:
Our example has an average count of four occurrences in this given period of time, but if this were a true Poisson process, zero calls would occur about 2% of the time, and both three or four call occurrences would hit you just under 20% of the time, with larger “hits” trailing off toward a near-zero chance of occurring. 
But what happens if we reduce the period of time we are reviewing to, say, every five minutes. If your average is consistent, you should receive an average of one call during that five-minute period. But in that short period of time, a true Poisson process looks even more “squashed” toward zero:
During about 37% of successive five-minute periods, you will receive no calls, and just one call for another 37% of the periods. About 6% of the time, you will receive three calls in that five-minute period, and a bit over one in one-hundred periods, you will get hit with four calls. Can you see where the perception of “clumping” comes in? 
Let’s look it another way. If the calls come in evenly spaced over that twenty minutes, it would look something like this, giving you five minutes to answer each call:
But it might look like this, a pattern that still has a rate of four calls every twenty minutes. Now you are in trouble:
Because of the nature of averages when time is concerned, just one single long wait skews the average interval even when there are many short waits. When we encounter the shorter time intervals after a longer wait, our minds perceive those most recent calls as “squeezed together” in time. This is the phenomenon called Poisson clumping.
If we were to compute the average weight of all of the mammals in a zoo, it would be skewed a bit high, because it takes, for instance, well over 100,000 mice to offset the weight of one elephant. So it is also with averaging time.
The number three in a “clump” is arbitrary. We typically don’t perceive a pattern after just two, and the probability of Poisson clumps bigger than three usually declines quickly, as the illustration above showed.
People often cite the “things happen in threes” phenomenon as if it were some sort of mystical or supernatural experience. I hate to pop this bubble, but most of the time it just didn’t happen as your brain constructed its story, and even in clean Poisson processes, it is just the math of “time-based averages.”
In our world, “stuff,” good or bad, often happens in clumps, and that is, oddly, “normal.”
- Why this pattern is true for all Poisson processes with an average of four is a fascinating story (if mathematics can have fascinating stories) that will wait for another day.
- Note that we are ignoring natural “crunch time” events, which complicate the calculations, but could be adjusted for. True Poisson processes are “independent” events like fair coin tosses. Even if I toss five “heads” in a row, the odds of tossing another “head” remain at 50%. One independent event, a phone call in this case, has no effect on the next one occurring.