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If you are a teacher then please also visit my new site: intermathematics.com for over 2000+ pdf pages of resources for teaching IB maths!

Amanda Knox and Bad Maths in Courts

This post is inspired by the recent BBC News article, “Amanda Knox and Bad Maths in Courts.”   The article highlights the importance of good mathematical understanding when handling probabilities – and how mistakes by judges and juries can sometimes lead to miscarriages of justice.

A scenario to give to students:

A murder scene is found with two types of blood – that of the victim and that of the murderer.  As luck would have it, the unidentified blood has an incredibly rare blood disorder, only found in 1 in every million men.  The capital and surrounding areas have a population of 20 million – and the police are sure the murderer is from the capital.   The police have already started cataloging all citizens’ blood types for their new super crime-database.  They already have nearly 1 million male samples in there – and bingo – one man, Mr XY, is a match.  He is promptly marched off to trial, there is no other evidence, but the jury are told that the odds are 1 in a million that he is innocent.  He is duly convicted.   The question is, how likely is it that he did not commit this crime? 

Answer:

We can be around 90% confident that he did not commit this crime.  Assuming that there are approximately 10 million men in the capital, then were everyone cataloged on the database we would have on average 10 positive matches.  Given that there is no other evidence, it is therefore likely that he is only a 1 in 10 chance of being guilty.  Even though P(Fail Test/Innocent) = 1/1,000,000,  P(Innocent/Fail test) = 9/10.

Amanda Knox

Eighteen months ago, Amanda Knox and Raffaele Sollecito, who were previously convicted of the murder of British exchange student Meredith Kercher, were acquitted.  The judge at the time ruled out re-testing a tiny DNA sample found at the scene, stating that, “The sum of the two results, both unreliable… cannot give a reliable result.”

This logic however, whilst intuitive is not mathematically correct.   As explained by mathematician Coralie Colmez in the BBC News article, by repeating relatively unreliable tests we can make them more reliable – the larger the pooled sample size, the more confident we can be in the result.

sally clark

Sally Clark

One of the most (in)famous examples of bad maths in the court room is that of Sally Clark – who was convicted of the murder of her two sons in 1999.  It has been described as, “one of the great miscarriages of justice in modern British legal history.”  Both of Sally Clark’s children died from cot-death whilst still babies.  Soon afterwards she was arrested for murder.  The case was based on a seemingly incontrovertible statistic – that the chance of 2 children from the same family dying from cot-death was 1 in 73 million.  Experts testified to this, the jury were suitably convinced and she was convicted.

The crux of the prosecutor’s case was that it was so statistically unlikely that this had happened by chance, that she must have killed her children.  However, this was bad maths – which led to an innocent woman being jailed for four years before her eventual acquittal.

Independent Events

The 1 in 73 million figure was arrived at by simply looking at the probability of a single cot-death (1 in 8500 ) and then squaring it – because it had happened twice.  However, this method only works if both events are independent – and in this case they clearly weren’t.  Any biological or social factors which contribute to the death of a child due to cot-death will also mean that another sibling is also at elevated risk.

Prosecutor’s Fallacy

Additionally this figure was presented in a way which is known as the “prosecutor’s fallacy” – the 1 in 73 million figure (even if correct) didn’t represent the probability of Sally Clark’s innocence, because it should have been compared against the probability of guilt for a double homicide.   In other words, the probability of a false positive is not the same as the probability of innocence.  In mathematical language, P(Fail Test/Innocent) is not equal to P(Innocent/Fail test).

Subsequent analysis of the Sally Clark case by a mathematics professor concluded that rather than having a 1 in 73 million chance of being innocent, actually it was about 4-10 times more likely this was due to natural causes rather than murder.  Quite a big turnaround – and evidence of why understanding statistics is so important in the courts.

This topic has also been highlighted recently by the excellent ToK website, Lancaster School ToK.

If you enjoyed this topic you might also like:

Benford’s Law – Using Maths to Catch Fraudsters

The Mathematics of Cons – Pyramid Selling

Essential resources for IB students:

1) Revision Village

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Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.  I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

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There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions.  What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

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The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers.   These all have worked solutions and allow you to focus on specific topics or start general revision.  This also has some excellent challenging questions for those students aiming for 6s and 7s.


Essential Resources for IB Teachers

1) Intermathematics.com

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If you are a teacher then please also visit my new site.  This has been designed specifically for teachers of mathematics at international schools.  The content now includes over 2000 pages of pdf content for the entire SL and HL Analysis syllabus and also the SL Applications syllabus.  Some of the content includes:

  1. Original pdf worksheets (with full worked solutions) designed to cover all the syllabus topics.  These make great homework sheets or in class worksheets – and are each designed to last between 40 minutes and 1 hour.
  2. Original Paper 3 investigations (with full worked solutions) to develop investigative techniques and support both the exploration and the Paper 3 examination.
  3. Over 150 pages of Coursework Guides to introduce students to the essentials behind getting an excellent mark on their exploration coursework.
  4. A large number of enrichment activities such as treasure hunts, quizzes, investigations, Desmos explorations, Python coding and more – to engage IB learners in the course.

There is also a lot more.  I think this could save teachers 200+ hours of preparation time in delivering an IB maths course – so it should be well worth exploring!

Essential Resources for both IB teachers and IB students

1) Exploration Guides and Paper 3 Resources

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I’ve put together a 168 page Super Exploration Guide to talk students and teachers through all aspects of producing an excellent coursework submission.  Students always make the same mistakes when doing their coursework – get the inside track from an IB moderator!  I have also made Paper 3 packs for HL Analysis and also Applications students to help prepare for their Paper 3 exams.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.