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.
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).