About 100 year floods (and probabilities, and risk analysis)

I'm based in Boulder, and it's flooding.  Luckily, we're all fine here.

Flooding is kind of a weird natural disaster.  First: It doesn't feel very threatening: It's just raining outside... a lot. But it does post some serious risks and deserves to be taken seriously.

100 year floods (Should be called: the 1% flood)

Near Boulder Creek at 7:11 AM on 9/13/2013
Near Boulder Creek at 7:11 AM on 9/13/2013

When engineers discuss risk analysis, we like to rank things according to their return periods (i.e., how long is our design really relevant for).  The common design parameter is the 100-year flood, which has an unfortunate name because people assume the 100-year flood happens roughly every 100 years.  This leads to statements like "The last 100 year flood was in 1894, so we're 19 years overdue for another one."

Really, a better name for the 100-year flood would be the 1% flood, in that each year there's a 1% chance of having the "100-year flood".  Ranking the floods assumes that year to year is uncorrelated, so each year it's like rolling the dice and you have a 1/100 chance of getting a flood that big.

Throwing Dice

chance of rolling at least one 6 in x throws.
chance of rolling at least one 6 in x throws.

If you have a 6 sided die, and you roll it 6 times, you're not guaranteed to get a 6 any of those times.   In fact, you've only got a 66.5% chance of getting a 6 in those 6 rolls (1-(5/6)^6).  These arn't bad odds... but it's not a "sure thing" either.

If you wanted a 90% chance of rolling at least one 6, you should roll the die 13 times.  If to you a "sure thing" was a 99% chance: you'd have to roll the dice 26 times, and even then, you have a 1% chance of still not rolling a 6 in 26 rolls.

But, you still have a 1/6 chance to roll a 6. Every time. If you got 100 people in a room, and had them roll dice and count how many throws it took them to get a 6, your average would be 6 throws.

Back to floods

So back to the floods.  In any given 100-years, it's really only 63% (1-(99/100)^100) likely that you'll experience a 100-year flood.  That said, it is also 0.01% possible to experience two 100-year floods in 100 years.   This is why calling it "The 100-year flood" is a horrible idea.

It's been 119 years since the last "100-year flood" in Boulder.  The chances of not having a "100-year" flood in a 120-year period are 30%, which isn't really that weird at all.  And we are still under the assumption that next year, the chances of having a 100-year flood are 1/100.  Next year doesn't know about the past 120 years.

Water receding and the sky is clearing
Water receding and the sky is clearing

But wait: it gets worse...

See my next post about how to calculate the 100-year flood and some more problems with it's name.  These include problems like small data sets and non-stationarity, but also different ways of calculating the 100 year flood yield different results.