### The Earthquake Paradox: A Probability Puzzle

Today (April 18, 2006) is the 100th anniversary of 'The Great Quake' in the San Francisco Bay Area. This reminded me of a puzzle in probability that I came up with a while back (as far as I know it, this idea is original to me since I haven't heard of this before) that I call the 'Earthquake Paradox.'

I came up with the Earthquake Paradox when I came across the 24-hour major earthquake probability prediction map of California created by the U.S. Geological Survey (USGS) (http://pasadena.wr.usgs.gov/step/). According to the USGS, "there is a 62% probability of at least one magnitude 6.7 or greater quake, capable of causing widespread damage, striking somewhere in the San Francisco Bay region before 2032." The risk of a major earthquake, if anything, is probably being understated; most geologists' intuitions would tell them that it is almost certain that there will be a major earthquake in California in the next few years. At any rate, most geologists believe that it is almost certain that California will suffer a number of catastrophic earthquakes in the decades to come.

Yet, on most days, when you look at the probability prediction map for major earthquakes (major earthquakes being 5.0 or above on the Richter Scale and a 'VI' on the Modified Mercalli Index) in California, the chances of a major quake are quite low. For example, looking at the portion of the probability map for the San Francisco Bay Area, the probability of a major earthquake hitting San Francisco in the next 24 hours, on most days (most people would think of it as the 'average' range of probabilities), ranges from 0.0001 to 0.00001 (i.e., usually less than 1% chance within a 'typical' 24 hour period).

So here is the paradox: How can it be that, over the not too distant long run (which includes the possibility that the event could take place within the next few days), it is nearly certain that California will experience a major earthquake, but, on a 'typical' (or 'average' or 'most') day(s), the probability of a major earthquake happening is fairly small?

I don't want to get into all of the reasons why I think this 'Earthquake Paradox' exists due to the constraints of space and time on a blog like this, but let me give an extremely brief glimpse of my thoughts on this matter. I believe that the Earthquake Paradox exists because of both mathematical and psychological reasons.

The mathematical / statistical reason for the existence of this paradox is, in part, due to the idea that randomness and risk does not always conform to the so-called 'normal' (Gaussian) distribution that is commonly used in both the social and the natural sciences. If we find that -- in both nature and in human social dynamics -- probabilities work in 'wild' and discontinuous ways (with unpredictable 'jumps,' etc.) compared to the way we are typically taught that chance works, then the paradoxical situation of having long term near certainty with short term unlikelihood is plausible from a mathematical standpoint. By the way, this is reminiscent of what Benoit Mandelbrot and Nassim Nicholas Taleb have been preaching -- the distinction between "wild randomness" and "mild randomness" in finance -- recently.

The psychological explanation for the Earthquake Paradox (again very briefly due to constraints) is because people tend to discount (I'm using this word in a technical sense as it is used in finance, economics, and more quantitative branches of psychology) future values -- be they prices or probabilities. Simple discounting, as understood by economists and some psychologists, wouldn't adequately explain the paradox. However, if we can accept the idea that people tend to more drastically discount future values and possibilities than allowed for by typical financial theory (and other disciplines), then the paradox, again, begins to be plausible. I believe that humans often base their discounting of the future based on what can be loosely termed as a 'hyperbolic' function. For some strange reason, economists seem to think that using a hyperbolic function is somehow less 'rational' than using the utility discounting function that underlies standard models; from a purely mathematical or scientific perspective, using such a discounting function is not any less 'rational' than using the function preferred by economists.

One final note ... there is an interesting link between geology / geophysics and finance / econophysics. Didier Sornette, a geophysicist at UCLA, has done some interesting research on financial econophysics. Perhaps there is a greater overlap between these seemingly disparate fields than meets the eye.

I came up with the Earthquake Paradox when I came across the 24-hour major earthquake probability prediction map of California created by the U.S. Geological Survey (USGS) (http://pasadena.wr.usgs.gov/step/). According to the USGS, "there is a 62% probability of at least one magnitude 6.7 or greater quake, capable of causing widespread damage, striking somewhere in the San Francisco Bay region before 2032." The risk of a major earthquake, if anything, is probably being understated; most geologists' intuitions would tell them that it is almost certain that there will be a major earthquake in California in the next few years. At any rate, most geologists believe that it is almost certain that California will suffer a number of catastrophic earthquakes in the decades to come.

Yet, on most days, when you look at the probability prediction map for major earthquakes (major earthquakes being 5.0 or above on the Richter Scale and a 'VI' on the Modified Mercalli Index) in California, the chances of a major quake are quite low. For example, looking at the portion of the probability map for the San Francisco Bay Area, the probability of a major earthquake hitting San Francisco in the next 24 hours, on most days (most people would think of it as the 'average' range of probabilities), ranges from 0.0001 to 0.00001 (i.e., usually less than 1% chance within a 'typical' 24 hour period).

So here is the paradox: How can it be that, over the not too distant long run (which includes the possibility that the event could take place within the next few days), it is nearly certain that California will experience a major earthquake, but, on a 'typical' (or 'average' or 'most') day(s), the probability of a major earthquake happening is fairly small?

I don't want to get into all of the reasons why I think this 'Earthquake Paradox' exists due to the constraints of space and time on a blog like this, but let me give an extremely brief glimpse of my thoughts on this matter. I believe that the Earthquake Paradox exists because of both mathematical and psychological reasons.

The mathematical / statistical reason for the existence of this paradox is, in part, due to the idea that randomness and risk does not always conform to the so-called 'normal' (Gaussian) distribution that is commonly used in both the social and the natural sciences. If we find that -- in both nature and in human social dynamics -- probabilities work in 'wild' and discontinuous ways (with unpredictable 'jumps,' etc.) compared to the way we are typically taught that chance works, then the paradoxical situation of having long term near certainty with short term unlikelihood is plausible from a mathematical standpoint. By the way, this is reminiscent of what Benoit Mandelbrot and Nassim Nicholas Taleb have been preaching -- the distinction between "wild randomness" and "mild randomness" in finance -- recently.

The psychological explanation for the Earthquake Paradox (again very briefly due to constraints) is because people tend to discount (I'm using this word in a technical sense as it is used in finance, economics, and more quantitative branches of psychology) future values -- be they prices or probabilities. Simple discounting, as understood by economists and some psychologists, wouldn't adequately explain the paradox. However, if we can accept the idea that people tend to more drastically discount future values and possibilities than allowed for by typical financial theory (and other disciplines), then the paradox, again, begins to be plausible. I believe that humans often base their discounting of the future based on what can be loosely termed as a 'hyperbolic' function. For some strange reason, economists seem to think that using a hyperbolic function is somehow less 'rational' than using the utility discounting function that underlies standard models; from a purely mathematical or scientific perspective, using such a discounting function is not any less 'rational' than using the function preferred by economists.

One final note ... there is an interesting link between geology / geophysics and finance / econophysics. Didier Sornette, a geophysicist at UCLA, has done some interesting research on financial econophysics. Perhaps there is a greater overlap between these seemingly disparate fields than meets the eye.

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