Probability of Storms
Do Real Storms Follow the Math?
 Chapter 3's analysis of the likelihood of Hurricane Katrina (shown at left) depends on the statistical nature of the frequency of storms. The chapter asserts this statistical nature, but the reader has to take this on faith.
These additional notes provide the missing background.
There are two questions: First, do real storms follow the statistical pattern known as the geometric distribution as asserted in the text? Second, does this distribution produce the phenomena described?
For the answer to the first question, we'll turn to an analysis of hurricane data provided by the U.S. government. To explore the second question, we will look at a simulation model of the arrival of storms with the same 1/50 probability of occurring as used in the text for either a Category 4 or 5 storm hitting New Orleans.
Intervals Between Storms for Real Hurricanes The histogram at right summarizes the intervals between the landfalls of hurricanes of all strengths that struck the U.S. over the course of 154 years (1851 to 2004). It is clear that the overwhelming period between storms is short, and that the length of the interval falls off quickly.
This pattern fits that of an geometric distribution in which the likelihood of a storm in any year is independent of the occurrence of previous storms. (Note that the most common interval is <1 year, meaning that more than one storm occurs in the same year.)
Independence of storms is a simplification, since it ignores hurricane cycles, climate change, and short term weather patterns. Still, the data fit the idealized model reasonably well.
Ideally, we would want this data to cover just Category 3, 4, and 5 storms, since it does not automatically follow that the interval between strong storms follows the same pattern as that of weaker storms (although it is true, since the forces that generate them are the same). Unfortunately for our analysis (but good news in human terms), stronger storms are much less frequent. As you will see in a moment, when the statistical sample is small, the interval pattern is harder to discern. A Simulation ExperimentIf we accept that occurrence of a storm in any year is independent of other storms, it is equivalent to a classical statistical representation. Imagine a bag filled with 100 balls, 98 of which are white and two of which are red. If we repeatedly reach into the bag and withdraw a ball, replace it, and keep doing so until we withdraw a red one, we can track the number of draws required until the red ball is drawn. This corresponds to the interval between storms using a probability of a storm in any single year of 1/50.
In the histograms below, a simulation of 100,000 runs is summarized. That is, the computer replicates the draws from the bag, stopping each time a red ball is drawn. Recording the number of draws needed, the simulation then moves on the the next run. To match the intervals between actual storms, as shown above, the number of simulated draws is reduced by one. In other words, if a red ball were to be drawn on the first attempt, that is recorded as a run length of zero.
You can see that in a short series of 10 runs (upper left histogram), more than half the intervals are very short -- less than 25 periods, or half the average period of 50. As the number of simulation runs rises, the histogram smoothes out and approaches the theoretical distribution. For the final simulation of 100,000 runs, the maximum interval is 579 periods (not visible on the chart). This corresponds to nearly 600 years passing between a Category 4 or 5 storm hitting New Orleans! While such an event is possible (since any interval is possible), its likelihood is very low. Short intervals, in contrast, are the norm. Let me know if you're convinced.
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