Friday, July 6, 2012

Century storms

"That's two straight years we've had a 'storm of the century'; those weather guys are idiots!"  Not long after I moved to the Washington, DC area, this happened, and the quote is real.  I'm not sure that the storms involved really were 'storm of the century' events -- events that if we had a long enough record, we'd see happen about 10 times per 1000 years -- but it's something to think of a little quantitatively, particularly in light of my normally abnormal note.

Let's suppose that we're building a house and would like it to last 30 years.  Well, to be specific, let's say we'd like a 99% chance of it lasting that long.  Obviously it has to be able to survive events that we'd expect to happen once per year.  And we can probably ignore things that we'd expect only once in a million years.  But what about a once in 100 year event?  The name misleads us in to thinking that the next time such an event would happen is 100 years after the last time.  While natural reading, it's wrong mathematics.  We could easily be in the unlucky 30 years that sees a 100 year event.  We could even see it twice.  But is there less than a 1% chance of having one 100 year event in a span of 30 years?  That's our design requirement.  If it can be expected more often than that, our house design is not reliable enough.  We need something better. And we'll need to get quantitative.

To deal with situations like this, where we have an expected number of events in a time span, whether it's number of 100 year events in the 30 years we want our house to last, or number of times a phone will ring in the next hour given how many times it usually does, we use the Poisson distribution.  For the Poisson expectation value (lambda) take the number of years you want and divide it by the N in N-year event.  100 at the moment.  In 30 years, then, we expect to see 0.3 '100 year' events.  Of course you can't have 0.3 of an event, you have 0, 1, 2, ... events.  One of the things the Poisson distribution does is to make the translation from our expectation in to something that could happen.  The other, of course, is that it tells us how likely it is to happen.

NumberProbabilityCumulative
074.1%0
122.2%22.2%
23.3%25.5%
30.3%25.8%

More than a 1 in 4 chance (greater than 25% chance) of at least one 100 year storm hitting our house in 30 years!  If we want that confidence, only a 1% risk of our house being destroyed in 30 years, we need to plan not for 100 year storms (which give us 26 times that risk!) but for 3000 year storms!

I used the online calculator at Graphpad Software.  There are a huge number of online calculators, just search on Poisson distribution calculator.  I encourage you to play around with this some, for events you're interested in, and levels of confidence you consider reasonable.

I was surprised at the results here.  One thing it suggests, as my house is more than 30 years old, is that we are already doing some design and construction to these levels.  It isn't a radical new concept.  The other is, when we are considering planning against catastrophic failure, we have to consider extremely rare events.  Most of us expect to live more than 30 years.  The related thing, just ask insurance companies, is that changes to the probabilities of extremely rare events are extremely important.  Changing the average temperature from 15 to 17 C (60 to 63 F) might be ignorable or even welcome, depending on who you are.  But turning the extremely rare 100 year storm in to a rare 10 year storm gives you only a 5% chance of your 'storm of the century' house surviving 30 years. (And instead of 0.4% chance of 3 or more such storms hitting you, it becomes 58%!)

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