Sunday, September 7, 2014

Casualties of modern society mathematical illiteracy:

Lucia de Berk, Colin Norris, Ben Geen, Susan Nelles

A good friend of mine, Michael Slott, once made the following observation: claiming illiteracy carries a social stigma, but stating that "I don't know math" is perfectly acceptable. Granted, math deals with abstractions you don't really learn until you assimilate the content and pure memorization is not enough.  But given the proper motivation, it is not that hard to understand it.

In his last interview, Carl Sagan noted: "people read the stock market quotations, and financial papers." Also: "people are able to look at sports statistics. look how many people can do that."

The trouble comes when emotions enter the picture and under the right circumstances it can lead to really bad consequences. 

Statistics can be counter-intuitive. For example, rolling the dice and getting six times in a row 1-1, for the seventh time you may think you are in a lucky streak and the chance to get the same thing is really high. Casinos know very well how to take advantage of our emotions and lack of statistical intuition, and so are the state lottery systems. 

Now in statistics there is a common mistake: the prosecutor's fallacy. To explain this we first need to explain Bayes' theorem:

\( P(A|B) = \frac{P(B|A) P(A)}{P(B)} \)

The simplest way to understand this is to derive it as follows:

P(A|B) is the probability to have A given B
P(B|A) is the probability to have B given A


P(A|B) * P(B) = the probability to get both A and B = \( P(A \cap B) \) 


P(B|A) * P(A) = the probability to get both B and A = \( P(B \cap A) \)

Because \( P(A \cap B)  = P(B \cap A)\) we have:  P(A|B) * P(B) = P(B|A) * P(A) q.e.d.

Now let A be the evidence (E) of guilt, and B the innocence (I). Prosecutor's fallacy is ignoring Bayes' theorem and concluding that:

P(I|E) is small because P(E|I) is small

This is best illustrated with a lottery example. Winning the lottery has a very small probability, but does winning it make you a cheater? Prosecutor's fallacy implies that:  

P(honest person given that you won the lottery) = P(honest | won the lottery) = P(winning the lottery | honest person) = P(winning the lottery by a honest person) = tiny percentage

Therefore you are a cheater.

Now onto Lucia de B case. She was a nurse in a children's hospital and some of the kids there died while under her care. Some doctors called the police to investigate and she was ultimately convicted based on the Prosecutor's fallacy which yielded a 1 in 342 million chances that a nurse is present by chance at the scene for the unexplained deaths . In fact the correct calculation was 1 in 25.  Fortunately the conviction was ultimately overturned, but other similar cases were subsequently discovered.   

In Netherlands there was Lucia de Berk, in UK Colin Noris and Ben Geen, in Canada Susan Nelles

Are there any unknown US cases?

So how do this happen? In a stressful hospital situation something out of the ordinary occurs. Then someone who stands out from the crowd is associated with the incidents. 

The situation snowballs and all past unexplained events are associated in an "aha moment" with the "serial killer". 

Side note: I recall a personal experience with this pattern matching psychological effect. In 2002 in Washington DC area there were two snipers who randomly shot people. Police were clueless and at some point they announced on the radio that the snipers were firing from a white van. The next day I was amazed to realize how many white vans were on the road, and probably all the white vans were pulled over by the police that weak. In the end it turned up that the white van information was incorrect. Now if you did not experience this kind of police drama, recall the last time you bought a new car. You tend to notice so many cars on the road identical with yours.

The case goes to trial and faulty prosecutor reasoning is not properly countered by the defense due to sheer math ignorance and/or incompetence. The "serial killer" is unfairly convicted and only the dedication of honest experts and the public pressure can sometimes overturn the conviction. 

When emotions run high, logic takes a back seat. In US there was a racially motivated nurse cases, in France there was the famous Dreyfus case. We should "J'accuse" elementary mathematical ignorance. 


  1. Thanks for the mention. I am proud to be your good friend. How about the Simpson's paradox? ... closely related, I think.

  2. Hi Mike, thanks for the kind words, I am lucky to have a friend like you. Simpson's paradox is counterintuitive, but there is no mystery there. On Wikipedia in the upper right corner there is a nice picture which explains it: think of a rising linear function and at some point have a discontinuity dropping the value followed by the same slope rising function. On each continuous section the trend is positive, but overall the average slope is negative.

    I just uploaded the Sleeping Beauty problem which is still open. Personally I am in the "thirder" camp. The Monty Hall problem is completely understood, but the surprise there is that well qualified people, including a Nobel prize winner insist on the wrong solution. I ran into this myself, and no matter what arguments I gave I could not convince the other side I was right.