# Statistical Likelihood of Extreme Events and the Ellicott City Floods

### Wednesday August 10, 2016

###### •  Earth science • environmental policy •  environmental science •  environmental studies •  flood studies •  Howard County •  mathematics •  risk management •  statistics •

This time, the flood hit close to home, literally. Over the last week, Ellicott City has responded after a very bad storm dropped nearly six inches of rain in only two hours. The recovery effort will be long and difficult, but the Ellicott City Partnership has done a remarkable job coordinating resources for both immediate and long term needs. The storm and its effects were severe and Ellicott City will require strong leadership, going forward.

However, there’s been some accusations and finger pointing, about who may be responsible for inflicting what damage. The evidence given is usually the gross severity of this storm and the remnants of Tropical Storm Lee in 2011 are just too close together. Something must have happened to cause this:

You can’t have a thousand year storm every five years.

Well, yes, you can. I am not defending the practices of those accused, but as evidence goes, this means nothing. As usual, people have misunderstood risk, something I’ve written about so often here, it has it’s own category. Let’s look at the numbers.

First, let’s talk about what it means to be a one-hundred year event or a one-thousand year event. It does not mean it will only happen once in a hundred years or once in a thousand years. It means the event has a 1/100 or 1/1000 probability of happening any given year. And, and this is the interesting thing, it also only has a probability of happening 1/100 or 1/1000 in back-to-back years. It seems natural to say “The probability is 1/100 this year and 1/100 next year, therefore the probability is 1/10000 of getting back-to-back one-hundred year events.” While the first part is true, the second isn’t. The second assumes the year pair is predetermined. Only the second is; the first can be any year. Therefore, we don’t start counting probability until the first one even happens.

Because of this, it is suddenly possible to see the probability of observing two one-hundred year events in five years. We don’t count the first again, then it is the probability of the event happening in the next four years. Again, it seems natural to say, “Well, it would be $1/100 * 4 = 4/100 = 0.04$.” And while close, it is not correct. It is, in fact, 1 minus the probability of it not happening in the next four years. That’s not terribly intuitive, but it is correct. So that is,

$1 -.99^4 \approx 1 - 0.96059601 = 0.03940399\text{,}$

which is almost, but not quite 0.04. We’ll make the math simpler and call it at 4% anyway. That’s the probability of observing two one-hundred year events within a five year period. One in 25.

But wait, there’s more.

If we want to know, what is the probability of it happening in downtown Ellicott City, it is 4%. And, understandably, Ellicott City residents feel like they are on the short end of a vicious stick. If the 2011 and 2016 storms had, instead, caused this kind of devastation in Savage, Savage residents would be asking the same questions. But Savage wasn’t destroyed and nobody there is asking.

So what if we wanted to know if it would happen in any community in Howard County. Well, community is very poorly defined, and there are no cities or other formal boundaries (except HOA lines) in Howard County, so let’s find a good proxy. We will use the Census-designated place (CDP). I kind of think CDP is too big, since we are talking about a highly-localized phenomenon, but we will use CDP since it is a good external standard. All of the following CDPs are located at least partially in Howard County:

• Columbia
• Elkridge
• Ellicott City
• Fulton
• Highland
• Ilchester
• Jessup
• Mount Airy
• North Laurel
• Savage
• Scaggsville

The probability of two one-hundred year events in five years happening in any of these 11 CDPs is, again, one minus the probability of it not happening. So that is

$1 - .96^{11} \approx 1 - 0.63823933055 = 0.36176066944\text{,}$

just about 36%. There’s some round-off error here, and the real number is closer to 35%, but that’s not a big deal right now. What we learn from this is the probability of any year being the start of two one-hundred year events in five years, in any CDP in Howard County, is more than one-third.

In simpler terms, yes, you can have a one-hundred year storm every five years, and almost certainly some place will. And this analysis only covers Howard County. Maryland or nationwide, the probability skyrockets. It’s rather like playing the lottery. Just because you probably won’t win doesn’t mean nobody will.

While I am unconvinced that this was a thousand-year event (but read the NOAA analysis), recalculating these probabilities for one-thousand year events is a trivial exercise left to the reader.

Risk is difficult to measure and difficult to grasp. Further, risk needs to be assessed holistically by policymakers and planners. There are certainly aspects of Main Street that could be redone to better accommodate a massive storm. But to assume that these events are so terrible as to be unnatural is really not grounded in the numbers.

Image by Scott Saghirian / Wikimedia Commons.