Tuesday, May 3, 2011

The likelihood of rare events

Having reviewed the geometric distribution while TAing a recent stats class, I thought I'd apply it to viewing the likelihood of extremely rare events. Sparing you the math, this tool tells us the likelihood of an event occurring over several chances, based on the likelihood of it happening with a single chance. The first figure looks at a simple series of coin tosses. The more times you play, the less chance one has of just getting either heads or tales. It also works with rare events. So, if you say some disaster (or whatever) only occurs 1 out of 1000 years, the second figure will use math to explain the chance of it having happened over multiple years. The figure is based on the assumption that time starts after the last event of this kind, and it's in the form of a cumulative distribution function.

Take a 1000 year flood for example. The earth doesn't have a memory as such* and so even just 10 years after such an event there is still a small chance that it will have happened again. You'll also notice that after 1000 years it's not assured that the event will have occurred (cause it's only 1 in 1000 on average); even after 1500 years it only has a 77.7% chance of having happened.

The scary thing is the first part of the figure. By year 223 there's a 20% chance that the event has already occurred, and it's a 1 in 1000 yr event! So whether we're talking earthquakes, end of civilizations, floods, droughts, or whatever, these
events can happen much more quickly than we anticipate. Do you think that any given country lasts 1000 years on average? Or that that's the frequency of devastating earthquakes? Well if it hasn't occurred around here for over 223 years (age of USA, anyone?) then there's a 1 in 5 chance of it having happened already (considering all of the alternative histories that could have played out).

One of the biggest planning mistakes people make is to treat the unlikely as impossible.^

*Technically, many environmental variables due show small autocorrelation (i.e., memory), but it depends on what we're dealing with. The zero autocorrelation assumed above is a reasonable simplification, since we don't have enough data on rare events to meaningfully estimate this complication. Higher autocorrelation would make the events cluster in time (as with the earthquakes we've seen in Japan).

^While he's unlikely the first, Larry Swedroe makes this point in his book and in the bogleheads forum.

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