Feb 01

Once in a Hundred Years

On the train home today, I grabbed an mX and started reading it. It wasn't until I got to the Letters to the Editor page that I saw this little gem. It made me realise how little people actually know about probabilities.

Letter in the mX today (2013-02-01)

Let's start with the basics: a "once-in-a-hundred-year flood" has a probability of occurring in any given year of 1% or, another way, the chance of not flooding is 99%.

The formula for working out the probability of having exactly a certain number of floods (n) within 100 years is:

So the chance of not having any large floods within a 100-year period is 36.6%

See the table below for a more complete list of flood probabilities:

$n$ $P(n)$ $P(\ge n)$
0 36.6% 100%
1 37.0% 63.4%
2 18.5% 26.4%
3 6.1% 7.9%
4 1.5% 1.8%
5 0.3% 0.3%
6 0.05% 0.05%

We can expand this formula to answer the question postulated in the letter above. The possibility of have 2 floods in 3 years is only 0.03% ( $\left(\frac{1}{100}\right)^2\times\left(\frac{99}{100}\right)^1\times{^3C_2}$ ). This has a chance of happening once every 3367 years. Although we don't know how many floods will come in the years ahead.

Of course, there is the possibility that the original calculations were wrong and these floods should happen more often naturally. Otherwise there's a 26.4% chance that we'll have more than 2 floods this century which are a lot higher than one-in-a-hundred to most people.

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