Estimating the Probability a Flood Occurs over the Length of Your Mortgage
Published:
Calculating Flood Risk
If at some point you ever want to buy a house near water, a variation of this question will undoubtedly pass through your head:
What are the chances my {insert name of your expensive piece of property close to water} floods?
In this post we will calculate the chance of an event happening, ie. the chance your beautiful riverside property floods over the life of your 30-year mortgage. We will do this using our understanding of probability and statistics. The result will be a pretty figure and useful table you can return to when you want to understand the likelihood of something happening. As a tl;dr, I’ll put them here at the top too:
Laying the groundwork
In my experience the question of understanding the probability of flood exceedance is a great example of where calculating the probability of something not happening is more intuitive and easier to understand. It also shows how flood terminology obscures/confuses the probabilistic nature of estimating a chance of flooding.
Flood Events, Return Periods, and Probability
I have found that there is a fundamental misunderstanding of the vocabulary used to communicate the probability of a flood event. You may have heard something like this before:
“Bob, the heavy rains north of Waterville were responsible for the 100-year flood event downtown.”
or
“The city updated their flood maps to include our house in the 100-year floodplain”
Do you know what is really meant by the term “100-year Flood”? It is NOT the case that this event will only happen once every 100 years.
Definition: a “X-year Flood” conveys that in any given year there is a $\frac{1}{X}$ chance that a flood event of that magnitude will occur. The name 100-year Flood conveys that there is a $\frac{1}{100}$, or 0.01 (1%,) chance of that magnitude event occurring; for a 50-year flood, 0.02 (2%); 2-year flood, 0.5 (50%.)
Of course, all of these probabilities describe the chance an event happens in a single year. What is more helpful, is knowing the probability that an event occurs at least once over multiple years. Let’s calculate that.
Formulating Exceedance Probability
To make this calculation tractable, we first make some assumptions. (This is not just me making these assumptions, a lot of engineering design calculations are based upon these assumptions.) We assume that the probabilities of flooding each year are independent and stationary. Independence means that the probability of occurrence of a certain size flood in one year does not change the probability of a flood in any other years. Stationary means that the probability does not change over time; i.e. in 30 years the probability and size of a 100-year flood will be the same as today.
Next, we must reconsider the question: we want to know what the probability is of a certain sized flood happening at least once over $n$ consecutive years. The complement of this question would be:
What is the probability that a flood of size X does not occur during the next $n$ consecutive year?
In any given year it is 100% likely that it will or will not flood to a certain extent. Therefore, the probability that a flood of size X does not flood is the complement of the probability that it will, or
\[P(X > x) = 1 - P(X < x)\]Because the No-Flood condition, $P(X < x)$, for each year is independent of each other, their joint probability is their product. Thus, the probability that a flood of size X does not occur during $n$ consecutive years is
\[P(X > x)_{n} = (1 - P(X < x))_{1} \times (1 - P(X < x))_{2} \times \cdots (1 - P(X < x))_{n}\]or,
\[P(X > x)_{n} = \left(1 - P(X < x)\right)^{n}\]We now have a formula to calculate the probability that a flood of size X will not occur over $n$ consecutive years. Now we can take its complement to determine the probability that a flood of size X will occur at least once over $n$ consecutive years.
\[P(X < x)_{n} = 1 - \left(1 - P(X < x)\right)^{n}\]Calculating Exceedance Probabilities
Equation in hand, we can plug in any combination values we are interested in! The table below shows the probability that a flood of various sizes will occur at least once over a 30 year period.
The graph below shows how these probabilities grow over the course of 30 years.
You now can determine the chances your house will flood over any length of time for any size flood. This, however, will not help you build a castle on a swamp. For that, you will need to consult this guy.
Footnote: You may be asking, how does one estimate the flood extent for an event with 1% chance of happening every year? Great question. I might write about that in the future. Stay tuned.