In The Better Angels of Our Nature, I came across an interesting probability paradox. It is quite counterintuitive.
We can start thinking about this paradox with the following question:
Suppose you live in a place where lightning can strike at any point during the year. Lightning strikes are completely random—
Your house has the same chance of being struck by lightning every day, and the strike rate is once a month.
Now, your house was struck by lightning today, Monday. Which day is most likely to be the next day it is struck? Or, put another way: which day is most likely to be the day of the second lightning strike?
If your answer is “any day after today” (because the probability of a lightning strike is completely random), congratulations—you got it wrong.
The correct answer is “tomorrow,” Tuesday.
Why?
Let’s run through the calculation.
First, let’s roughly estimate the chance of being struck on Tuesday as 0.03 (once a month, remember).
Next, let’s look at the chance of being struck on Wednesday. For Wednesday to be “the next day the house is struck,” two conditions have to be met:
First, lightning must strike your house on Wednesday, which has a probability of 0.03. Second, lightning must not strike your house on Tuesday; otherwise Tuesday would be the day of the second strike, not Wednesday. So we multiply the probability that lightning does not strike the house on Tuesday (that is, 1-0.03=0.97) by the probability that lightning strikes the house on Wednesday (0.03). The result is 0.0291, slightly lower than the probability of being struck on Tuesday.
What about Thursday? If Thursday is the day of the second lightning strike, the probability is: Tuesday is not a strike (0.97), Wednesday is not a strike (also 0.97), and Thursday must be a strike, so the chance is 0.97×0.97×0.03=0.0282.
And Friday? The chance that Friday becomes the day of the second lightning strike is 0.97×0.97×0.97×0.03=0.0274.
As we move further into the future, the probability trends downward (0.03, 0.0291, 0.0282, 0.0274); the farther out we go, the lower it gets. More precisely, the probability decreases exponentially. The chance of being struck a second time only after 30 days from today is just slightly above 0.01.
This probability model is called a “Poisson process.” One important fact revealed by the Poisson process is:
Although the two events (the two lightning strikes) occur completely randomly and independently, the intervals between them (time intervals in this lightning example) are not uniformly distributed:
There are many short intervals, meaning consecutive days with lightning strikes are quite common because the probability of that happening is high; long intervals are rare because the probability of being struck again after 30 days is small.
In other words: even events that occur completely randomly can look clustered.
To put it even more bluntly:
Random does not mean average. In fact, quite the opposite: after a random event is repeated many times, it will inevitably produce a non-uniform distribution.
Randomness not being average is a deeply counterintuitive fact. Most people simply fail to recognize it.
During the London Blitz in World War II, Londoners noticed that several areas of the city were hit repeatedly by German V-2 rockets, while other areas were completely untouched. They believed the rockets were targeted at specific communities. In the minds of Londoners, if the rocket impact points were random, they should be evenly distributed across the whole of London.
But when statisticians divided a map of London into small squares and counted the number of rocket strikes, they found that the strikes fit a Poisson-process distribution—which is to say, the impact points really were random.
Londoners were probably overthinking it. At that time, Hitler’s Germany still had no precision-guidance technology; otherwise, there really might not have been a later “wrath of the Führer.”
References:
Steven Pinker (2015). The Better Angels of Our Nature. CITIC Press.
Note
- This article in the “Psychology New Knowledge Mini-Lessons” series was written to accompany the related content of Zhichao Wei: New Knowledge in Psychology, a popular-science short essay published on the now-defunct platform “Fantuan.” The original link is no longer available.
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