Random Clustering

By Alex Beal
April 6, 2017

The Better Angels of Our Nature has a particularly clear explanation of why random events cluster. Our lives are ruled by Poisson, so I think there’s some wisdom to be gained here:

Suppose you live in a place that has a constant chance of being struck by lightning at any time throughout the year. Suppose that the strikes are random: every day the chance of a strike is the same, and the rate works out to one strike a month. Your house is hit by lightning today, Monday. What is the most likely day for the next bolt to strike your house?

The answer is “tomorrow,” Tuesday. That probability, to be sure, is not very high; let’s approximate it at 0.03 (about once a month). Now think about the chance that the next strike will be the day after tomorrow, Wednesday. For that to happen, two things have to take place. First lightning has to strike on Wednesday, a probability of 0.03. Second, lightning can’t have struck on Tuesday, or else Tuesday would have been the day of the next strike, not Wednesday. To calculate that probability, you have to multiply the chance that lightning will not strike on Tuesday (0.97, or 1 minus 0.03) by the chance that lightning will strike on Wednesday (0.03), which is 0.0291, a bit lower than Tuesday’s chances. What about Thursday? For that to be the day, lightning can’t have struck on Tuesday (0.97) or on Wednesday either (0.97 again) but it must strike on Thursday, so the chances are 0.97 × 0.97 × 0.03, which is 0.0282. What about Friday? It’s 0.97 × 0.97 × 0.97 × 0.03, or 0.274. With each day, the odds go down (0.0300 . . . 0.0291 . . . 0.0282 . . . 0.0274), because for a given day to be the next day that lightning strikes, all the previous days have to have been strike-free, and the more of these days there are, the lower the chances are that the streak will continue. To be exact, the probability goes down exponentially, accelerating at an accelerating rate. The chance that the next strike will be thirty days from today is 0.9729 × 0.03, barely more than 1 percent. […]

For reasons we have just seen, in a Poisson process the intervals between events are distributed exponentially: there are lots of short intervals and fewer and fewer of them as they get longer and longer. That implies that events that occur at random will seem to come in clusters, because it would take a nonrandom process to space them out.1

I’m a quarter of the way through this book, and I always look forward to picking it up. To say that it addresses more than statistical curiosities is an understatement. The breadth of Steven Pinker’s research is staggering, jumping from history, to game theory, to statistics, to psychology in the span of a few pages. This is the type of book I’ll still be mulling over months after putting it down.

  1. Pinker, S. (2011). The Better Angels of Our Nature Why Violence Has Declined. The Statistics of Deadly Quarrels, Part 1: The Timing of Wars. Paragraph 10.↩︎