The Analytic Atavar

Idiosyncratic Musings of a Retrograde Technophile

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Location: Chandler, Arizona, United States

Monday, December 01, 2008

The 100 Year Storm

We sometimes hear phrases like "a 100 year storm or flood" or a "500 year storm or flood" or a "once in 10,000 year asteroid strike" used and most understand them to mean that such events will occur only once every 100 or 500 or 10,000 years. This is a misinterpretation of the statistics involved. The probability of a certain number of events in a certain time period can be calculated using the Binomial Theorem. Assume the annual probability of an event is 'p', so that the probability of it not occurring is (1-p). Then evaluate the following expression using the Binomial Theorem:
    [p + (1-p)]n = ∑ni=0 n!/[i!·(n-i)!] pi·(1-p)(n-i)
Each term represents the probability that exactly i events will occur in the n year period.

As a concrete example, consider a 100 year flood and a time period of 100 years. The annual probability of such an event is 0.01, and we take n = 100 for a 100 year period. Evaluating the first few terms of the Binomial Series:

i n!/[i!·(n-i)!] pi·(1-p)(n-i) E
0 0.366032341 0
1 0.369729637 0.369729637
2 0.184864818 0.369729637
3 0.060999165 0.182997497
4 0.014941714 0.059766859
5 0.002897787 0.014488935
6 0.00046345 0.002780704
7 0.000062863 0.000440044
8 0.000007381 0.000059053
... ... ...
Sum 0.999999156 0.999992366
The second column is the probability that exactly i events occur in a 100 year period, and the third column is the Expectation, which is the product of the numbers in the first two columns. As expected, the probabilities add up to 1 (neglecting the small probabilities for i > 8), and the expectation also adds up to one (again, neglecting the omitted terms), so we would expect an average of one event over many 100 year periods.

What is counter-intuitive is that there is almost as great a chance that no events occur (36.6%) as there is that 1 event occurs (37.0%), and since:
    1 - 0.366032341 - 0.369729637 = 0.264238022
there is a 26.4% chance that 2 or more events will occur in any 100 year period. It is therefore more likely that no events or 2 or more events will occur (63.0%) than that only 1 event will occur, a result most people would probably not believe !

Of course, this calculation assumes that the events from year to year are independent, but since weather seems to follow multiyear cycles, the independence from year to year is doubtful, and it seems likely that the probability of multiple events is higher than this calculation indicates.
P.S. - For those unable or unwilling to make such calculations, here is an on-line binomial calculator.


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