Calculated Risks

I’ve been reading Calculated risks, a book by Gerd Gigerenzer.  There is lots to make you think.

Here’s an example:

Your DNA matches a trace found on a victim of a crime.  The court calls an expert wetness who gives this testimony:

“the probability that this match has occurred by chance is 1 in 100,000.”

A chance match sounds very unlikely.  So does that mean you will be found guilty?  As Gigerenzer points out, the expert could have phrased the same information as:

“Out of every 100,000 people, one will show a match.”

Now we can see that, in a city like Melbourne, with a population of about 4 million people, about 40 people will show a match.  The point is, that the probability that you committed the crime is not the same as the probability of a match.  The probability that you committed the crime, given the DNA match, is only 1 in 40 if there is no other evidence and the potential perpetrators include anyone living in Melbourne.

Much of the book is about finding clearer ways of talking about, and thinking about, probability and risk.  An example:

The probability that a woman of age 40 has breast cancer is about 1 percent.  If she has breast cancer, the probability that she will test positive on a screening mammogram is about 90 percent.  If she does not have breast cancer, the probability that she will nevertheless test positive is 9 percent.  What are the chances that a woman who tests positive actually has breast cancer?

This is the way Gigerenzer presents the solution, using what he calls, natural frequencies.  Consider 10,000 women, 1% have cancer so that is 100 women.  Of these, 90% will return positive tests (i.e. 90 women with cancer will test positive).  Of the 9900 without cancer 9% will return positive test or 891 women. So there are 891 + 90 = 981 women with positive tests of which 90 have cancer.  So the chance that a woman with a positive test has cancer is 90/981, about 1 in 10.

Cancer.odg 2015-11-22 21-23-33

Different ways of getting a positive test result

We can grind through a problem like this with Bayes theorem but the natural frequency approach makes it easier to understand the problem intuitively.


Would the natural frquency idea help with the communication of hydrologic risks?  The latest guidance from Engineers Australia is that we should refer to the 1% annual exceedance probability (AEP) flood rather than the 100-year average recurrence interval (ARI) flood.  This moves away from expressing probabilities as natural frequencies but the rationale is that the ARI terminology is also confusing.  Many people think that only one “100 year flood” can occur every 100 years (see this forum about communicating flood risk in Christchurch, New Zealand)

When discussing flood risk, we could say something like: “Think of your house and 99 other houses, spread all over Australia, that have the same risk of flooding.  On average, one of these houses will flood every year”.

The risk is also the same as rolling a 100-sided die once a year.  If your number comes up, you get flooded.  On average your number will come up every 100 throws but it could come up any throw or several times in a row.

100-sided die

100-sided die (source)

Gigerenzer also talks about absolute risk reduction and relative risk reduction.

If a house is prone to flooding in a 1 in 10 year ARI event (a 10% AEP event) and we build a levee so it is now protected up to the 1 in 100 year ARI event (a 1% AEP event), then the absolute risk reduction is 0.1 – 0.01 = 10% – 1% = 9%.

The relative risk reduction is absolute risk reduction divided by the risk prior to treatment i.e. 0.09/0.1 = 90%.  So if you were to seeking to attract funds for a levee scheme, what sounds better: A 9% absolute risk reduction or a 90% relative risk reduction?

2 thoughts on “Calculated Risks

  1. lukecunningham33

    Hi Tony, great article. Reminds me of our recent work in the City of Melbourne together where we analysed 2 flood events (in the years 1972 and 2010) both looking like over 1 in 500 year events (or the 0.2% AEP). It will be interesting to see how the new IFD data changes things. To add confusion to the probability side of things in flooding, we often are faced with trying to explain that the 1 in 100 year ARI (1% AEP) rainfall event doesn’t necisarily result in a 1 in 100 year flood event. Or in conditions (say Melbourne CBD again) when we drop the 1 in 100 year ARI rainfall on the catchment (and assume it creates the 1 in 100 year flood) and apply the 1 in 10 year ARI Yarra River level. What’s the real ARI here? A 1 in 10 year event in the Yarra would be caused by rainfall in the upper catchment perhaps 3 days earlier or maybe a storm surge from the bay… are the events really linked or has the probability reduced again?

    1. tonyladson Post author

      I agree that these types of joint probably questions are very challenging because it is very difficult to work out how the different variables are correlated. Thinking about the CBD v Yarra River example, there would be some historical data. The Yarra is well gauged and CBD flooding would be reported in local media. I expect that CBD floods would be mainly caused by summer thunderstorms while Yarra flooding would be mainly caused by frontal rain in winter. If that were true, the correlation would be negative.

      The place that I would start to look is project 18 of the Australian Rainfall and Runoff revision.

      If I come across anything useful, I’ll post it.


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