"A 100-year flood" is one of the most misunderstood phrases in science. It does not mean a flood that comes once a century. It is a statement about chance: how often a river overflows its banks onto the floodplain it built, estimated from its record of discharge and the same probability as earthquake forecasts.
Why it works
Take the largest flow of each year for $n$ years and rank them, the biggest as $m = 1$. The recurrence interval of the flood ranked $m$ is estimated as $T = \dfrac{n + 1}{m}$: over a long record, a flood that big or bigger is matched or exceeded on average once in $T$ years. That makes its chance in any single year $p = \tfrac{1}{T}$. So a 100-year flood has a 1% chance every year, including the year after one happens. Over $n$ years, the chance of at least one is $1 - (1 - p)^n$: over a 30-year mortgage, about 26% for a 100-year flood.
Key idea
Recurrence interval $T = \dfrac{n + 1}{m}$ from ranked yearly peaks; yearly chance $= \tfrac{1}{T}$. A 100-year flood has a 1% chance every year.
Worked examples
Worked example: Rank a record
A town's records give 9 years of yearly peak flows. The second largest was 720 m³/s. What is its recurrence interval?
- $n = 9$, $m = 2$: $T = \dfrac{9 + 1}{2} = 5$ years.
- So a flow that large has about a 20% chance in any year. (A 9-year record is short; long records give better estimates.)
Answer: 5 years.
Worked example: Over a mortgage
What is the chance of at least one 100-year flood in 30 years?
- No such flood in a year: $0.99$.
- None in 30 years: $0.99^{30} \approx 0.74$.
- At least one: $1 - 0.74 = 0.26$, about 26%, as the USGS points out.
Answer: About 26%.
Common mistake
Many people think that after a 100-year flood, the next one is 100 years away. The river has no memory: the chance is 1% every year, and two can come in back-to-back years. Check the definition: 1 in 100 is a chance per year, not a schedule.