Estimating the probability of a major accident from the frequency of minor incidents

Major, catastrophic incidents (fortunately) do not occur very often. This means that few or no historical data are available on which a statistical calculation of probability can be based. On the other hand, similar but much smaller incidents may occur frequently. Can this data be used to estimate the chance that something large will happen? A typical shape of the probability distribution is as shown in the sidebar.

The data will allow us to estimate the probability of the more likely size incidents, i.e. around the peak of the curve, and the problem is to guess how the curve falls away at the high end. Statisticians will talk about “estimating the tail of the distribution”, or “rare event analysis”. One typical approach to this problem is to:

- Analyse the type of incident and derive a formula for the shape of the probability curve.
- Fit the curve to the available data.
- Use the fitted curve to calculate the probability of very large incidents.

This requires a lot of effort.However an initial rough estimate can often be obtained by the following reasoning. A big event is like many small events happening at once. Hence, boldly ignoring any dependencies, a first guess would be that, if the probability of an event of unit size is p, then the probability of an event of size S is P(S) = p**S. Or

P(S) = exp (- S)

So a first guess is that any such tail is exponential. This, conveniently, implies that the probability of an event larger than some size S is also exp (-S). “Unit size” is now determined by this same formula: it is the size such that the chance of events larger than this is exp (-1) which is roughly one-third. Hence, we can look at the recorded events and pick the level where two-thirds are below it, and one-third is above it.

If this type of reasoning is used, a lot can depend on how one chooses to measure “size”, and of course some data about the size of past events is needed. If we only have data that tell us how many events occurred and no information at all about how they differed in size or severity, then of course even this crude method cannot be used.

Thus it is important to not only record that an incident has happened, but to record a measure of its severity as well.

Whether you only want a rough idea of risks, or whether you need an accurate quantitative assessment, Novidec can help.

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