Over the holidays my dad posed a two-part question after dinner: If a vehicle goes 200,000 miles without a mechanical failure does that mean that it is more likely to have a failure soon? And does the same hold true for an employee who goes 25 years without a lost-time accident at work?
In answer to the first question I responded “no.” Failure rates of electrical and mechanical devices are typically modeled as an exponential distribution. This distribution is “memoryless” in the sense that the probability of having an accident at time t=1 given no accidents up til time t=0 is the same as the probability of having an accident at time t=2 given no accidents through time t=1.
For the employee question the answer is a bit more complex. Here I would assign workers to two types: accident prone or safe. The longer a worker is accident free, the higher the probability that they are in the safe category. (I would use a beta-binomial model with Bayesian updating for those playing at home). A more realistic analysis might include aging effects such as deterioration of eyesight and balance on the one hand, with gaining wisdom and on-the-job experience on the other. For all I know such a study may exist.
At its core this is the same idea as an athlete “coming due” in baseball–false–or having a “hot hand” in basketball”–also false. The interesting part about all this to me is that for two superficially similar problems different models are required by virtue of the fact that humans are involved. Mechanical parts are memoryless; people are not. That makes all the difference.