## “100 Year Flood Event”

Gene’s post on media reporting of the flooding is typical: Gene makes a confident claim, rob chimes in to meekly disagree, Gene bites rob’s head off, and we all end up wiser.

First and probably most important: The “100 year flood event” stuff is defined for a specific region. So over the country as a whole, we’d expect a bunch of “100 year flood events” every year.

(The odds of someone picking the winning lotto numbers make it way way less than a “once in a lifetime” event. And yet every couple of months, somebody wins the lotto. Thing must be rigged!)

But more interesting are the following distinctions:

==> A “100 year flood event” is defined as an event that has a 1% probability of happening in a given year.

==> This is NOT THE SAME THING as saying that we expect a flood of such magnitude to occur one time per century.

==> It IS equivalent to saying that the expected number of times of such a flood occurring per century is 1.

To understand the above statements, consider someone rolling a fair die. If we ask, “What’s the expected value of the roll?” we can calculate it by doing the probability of an outcome times the value of that outcome, and summing over all possible outcomes. So for our fair die, the expected value is 3.5.

Thus, the “expected value of the die roll” is 3.5.

But suppose I ask you, “Hey, do you expect the next die roll to turn up as 3.5?” then the answer is clearly no, since that is literally impossible. We can be quite sure the next die roll will NOT be 3.5.

The expected value of the die roll is 3.5, even though we do not expect the value of the die roll to be 3.5.

This all makes perfect sense to me, except for this part:

“==> This is NOT THE SAME THING as saying that we expect a flood of such magnitude to occur one time per century.

==> It IS equivalent to saying that the expected number of times of such a flood occurring per century is 1.”

I can’t find the difference between those two statements. “A flood of such magnitude is expected to occur one time per century” and “the expected number of times of such a flood occurring per century is 1” are definitely identical, yes?

Bob means that we do not expect that the flood will actually happen one time per century. Instead we mean that over a sufficiently long period the flood will

average outto happening one time per century.It makes a big difference for all those people who live less than one century and happen to be living through a rough patch.

To make matters worse in this particular case, our historic data doesn’t go all that far back, so we can’t produce a genuine average with a high degree of confidence… and anyway the climate does change (all by itself) and always has done (for example, a few hundred years ago was the “Little Ice Age”). Thus, the validity of taking a stochastic average inside a chaotic system is itself questionable. Lorenz worked on this problem, although his articles are harder to get these days, there used to be a long list available for free on his homepage.

“Bob means that we do not expect that the flood will actually happen one time per century. Instead we mean that over a sufficiently long period the flood will average out to happening one time per century.”

I’m still not sure how this is meaningfully different.

If someone held a gun to your head and asked you to estimate how many such floods would occur over the course of a century, “1” is still clearly the best answer, is it not?

Obviously it’s not

guaranteedto be 1, similar to how if you flip a coin 10 times, there’s no guarantee you’ll get 5 heads and 5 tails. But that’s still the smart bet.‘Obviously it’s not guaranteed to be 1’

I think that the heart of this issue – it seems there is a view that some people do indeed take ‘a 1 in a 100 year event’ to mean it will be guaranteed to be exactly 1 – and the media (according to the 538 article that Gene quotes in his post) think it worthwhile to refute this view.

This is possibly all about terminology but I find it confusing to say ‘It IS equivalent to saying that the expected number of times of such a flood occurring per century is 1’ when there is a >50 chance that there will not be exactly 1 flood. . or ‘The expected value of the die roll is 3.5’ when 3.5 cannot actually be rolled.

Wouldn’t it be clearer to say “the average number of times of such a flood occurring per century is 1” or “The average value of the die roll is 3.5” ?

And a quick google:

https://math.stackexchange.com/questions/904343/what-is-the-difference-between-average-and-expected-value

indicates that ‘average’ and ‘expected’ are interchangeable terms in statistics.

I’m going to do another post on this.

Imagine a “two year storm”. The chances in a given year of a storm is 1/2. What are the chances we’ll get such a storm in the next two years? There are 4 cases, all with chance 1/4: SS, SN, NS, NN. The chance of having a “two year storm” in the next two years is 3/4.

So you have to be careful exactly what you are asking. “How many two year storms do you expect in two years?” has the answer one. “How often should you expect a two year storm in the next two years?” has the answer 75 per cent.