The fourth school bus accident in the Richmond, Virginia area occurred this morning. Everyone wants to know, what does this mean?!?
Here’s what I think it means: bus accidents can be modeled as a Poisson process. Equivalently, the time between bus accidents can be modeled using the exponential distribution. This modeling paradigm is appropriate if bus accidents “randomly” occur independently of one another, which is a reasonable assumption.
If the time between bus accidents is exponentially distributed, then we expect that sometimes bus accidents occur in groups of three or four. Example exponential probability distributions are below. The exponential distribution has parameter lambda, where the average time between arrivals (bus accidents in this case). Most of the “meat” of the distribution is close to zero, even if the average time between arrivals is very large. This means that we would expect to sometimes observe small interarrival times and then go a long time between the next arrival.
Let’s put this in terms of bus accidents. If bus accidents occur as a result of chance or coincidence, then we would sometimes expect to observe four bus accidents in a week and then go months before the next bus accident. Four bus accidents in a week does not necessarily imply that something nefarious is going on.
This reasoning can also be used to explain why completely unrelated celebrity deaths sometimes occur in threes.
Example exponential distributions (probability density functions). The average time between arrivals is lambda^-1.
How rare are four bus accidents in a week? Let’s assume that bus accidents occur once every four weeks on average (lambda=1/4). The probability of observing 4+ accidents in a week is 0.01%. Pretty rare. But that’s any one week. The school year is 36 weeks long, which means that we would have 36 chances to have 4+ accidents in a week. Using the Binomial distribution, we find that the the odds of having at least one week with 4+ accidents is 0.5% (once every 200 years).
What about a slightly less extreme week? The probability of observing 3+ accidents in a week is 0.2%. Over the course of a year, the odds of having at least one week with 3+ accidents is 7.5% (once every 13 years).
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Punk Rock Operations Research 
April 6th, 2012 at 1:43 pm
“Accidents come in threes”. One of my colleagues used to point out that when you have observed a phenomenon such as “a school bus accident” then the most likely time to observe the next one was in the next time interval … another aspect of the Poisson process. And once there had been two such phenomena in quick succession, observers would be on the alert for further events. In addition, if you measure time from the first observation of a phenomenon, you are not selecting a random epoch as the start of the time interval.
April 6th, 2012 at 8:05 pm
So would medical ‘trauma’ be considered a Poisson process also? (trauma as in acute medical episodes of seemingly random and unrelated origin)
April 7th, 2012 at 9:12 am
Oh, stop it! You’re ruining a good conspiracy theory! What are people going to freak out about now?
April 7th, 2012 at 2:22 pm
@matforddavid: You are correct. A Poisson process has independent increments, and if we start to count with an incident (a bus accident in this case), then we are really interested in finding the probability of n-1 additional incidents in the next unit time (3 accidents in this case).
@Wipa: I think that trauma could be modeled as a Poisson process, particularly if one is modeling the number of people who enter a trauma center at a hospital. Trauma from a car accident likely occurs independently from trauma from falling off a roof across time. Mass trauma events (e.g., IED blasts in public places) would violate the Poisson process assumptions (although it perhaps could be modeled as a compound Poisson process).
@drmorr: There is plenty to freak out about! I hope to record a podcast about irrational fears. Stay tuned.
April 9th, 2012 at 11:15 pm
You notice that today Hospitals frequently post current wait times for Emergency Room arrivals. Isn’t this based on Poisson process assumptions?
April 16th, 2012 at 5:42 pm
Russ Stebner…
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