# Examples of Mathematics in Court

In court trials, natural sciences such as physics and biology routinely make an appearance, e.g. when estimating the speed of a vehicle based on impact damage or trying to deduce from the condition of a corpse how long ago the person died.

Pure mathematics is found in courtrooms much more rarely (perhaps because of the difficulty of convincing a judge – or worse, a jury – with mathematical arguments). I am currently aware of the following two broad categories of mathematics being invoked in a trial:

1. Probability and conditional probability, especially the misunderstanding thereof. One instance of this is known as the Prosecutor's fallacy and basically involves a confusion of prior and posterior in Bayesian inference. The Wikipedia page mentions the infamous "Sally Clark case" as a possible example of this fallacy resulting in a wrongful conviction. BBC News also has an article according to which mathematical reasoning about test results played a role in the Knox/Sollecito murder trial.

2. Benford's First-Digit Law which predicts the approximate frequency distribution of digits in real-life data, is not only used as a heuristic for the detection of possible fraud, but shockingly even accepted as evidence of the same by some jurisdictions.

Are there examples of other branches of mathematics that have played a role (for the better or worse) in the courtroom?

As a side note, is is interesting that much of the key courtroom vocabulary like "trial", "case", "law" etc. is also part of standard mathematics lingo, which makes meaningful web searches for such material quite challenging.

• matheducators.stackexchange.com/questions/5676/… Nov 14, 2014 at 12:10
• There's a section in Salvadori's Why Buildings Fall Down in which he describes testifying as an expert witness in a case involving a building collapse, explaining to the jury the equations that govern the maximum loading of a dome, and being cross-examined about that.
– MJD
Nov 14, 2014 at 14:28
• The infamous Indiana pi Bill comes to mind. :) Nov 21, 2014 at 10:23
• "Objection, your honor! The counselor is assuming the continuum hypothesis, which is clearly unprovable from the axioms of $\sf ZFC$ as shown by Goedel and Cohen!!!" $\tag*{}$ "Sustained." Nov 23, 2014 at 23:54

## 5 Answers

Although this falls into the category of Physics, this is a noteworthy example of beating a traffic ticket

http://arxiv.org/abs/1204.0162

• "The paper was awarded a special prize of $400 that the author did not have to pay to the state of California." :) – user139000 Nov 14, 2014 at 11:31 The book Math on Trial: How Numbers Get Used and Abused in the Courtroom by Leila Schneps and Coralie Colmez has many examples. B. Grofman and H. Scarrow, ‘Iannucci and Its Aftermath: The Application of the Banzhaf Index to Weighted Voting in the State of New York’, in Steven Brams, Andrew Schotter and Gerhard Schwodiauer (Eds.), Applied Game Theory, Physica-Verlag, Vienna,$1979$,$168$-$183\$, discusses the use and misuse of the Banzhaf power index by the courts; a PDF is available here.

Abstract of The use of trigonometry in bloodstain analysis:

Bloodstain pattern analysis (BPA) is a valid forensic method which belongs to the category of biological methods using trigonomic models. Despite its development through the years, the method has been re-formulated a standard one and globally used, recognized in standard sheets. This method permits exact analysis of the dynamic and characteristic properties of bloodstains after impact on surfaces such as floors, walls, and ceilings, various exterior and interior items, and clothes. It is also possible to determine the characteristics of blood from the outer part of the body. According to the presence of blood and its quantity, it is also possible to use this method for verification of reconstruction of criminal acts, while being tested for its validity with primary conditions of preserved and readable traces of blood. Even though this method is not considered as the major one or the only one information obtained in this way can be used for judicial. In our research, we tested the validity of this method in an experimental model using firearms. We compared measurements of the lengths of trajectory of impact and the height of the blood sprayed upwards from a distance of 1, 3, 5 and 10 meters. The experiment was based on two main presumptions. The first was the knowledge of the value of the distance and the angle of impact of the bloodstain, the second, the ability of the blood to reach a certain height and the angle of its impact. In accordance with trigonometric formulas, both the impact of the selected distance of drops of blood, and the height of the selected bloodstain could be determined without any verification of the flight trajectory and the distance of bloodstains. The results indicate that the method for these requirements differs from the real values, while increasing the measurements with the indicated spot of the shot. Aside from the unique values which were calculated, other results of the impact of the distance of drops of bloodstain were considered of lower value, and the values concerning the height of the bloods stains after the shot higher than real values. In spite of the lack of total accuracy, we recommend using this method widely and more often for investigation and verification of individual acts in criminal and forensic practice.

After the 2000 census, the state of Utah sued the Federal Government arguing that it should be given the delegate that went to the state of North Carolina. This document references some of the applied mathematics that went into making this decision. http://www.ncssm.edu/courses/math/Talks/PDFS/VotingAndPower.pdf