# A forensic formula for solving crimes

## That’s Maths: Forensic scientists need competence in mathematics to help solve crimes

Forensic biologists may have to work with crime scene samples that are small and degraded. This makes the identification of a unique individual more difficult, and probabilistic arguments are required to draw inferences. Photograph: Frank Miller

What use is maths? Why should we learn it? A forensic scientist could answer that virtually all the mathematics we learn at school is used to solve crimes. Forensic science considers physical evidence relating to criminal activity, and practitioners need competence in maths as well as in the physical, chemical and biological sciences.

Trigonometry, the measurement of triangles, is used in the analysis of blood spatter. The shape indicates the direction the blood has come from. The most probable scenario resulting in blood spatter on walls and floor can be reconstructed using trigonometric analysis. Such analysis can also determine whether the blood originated from a single source or multiple ones.

Suppose a body is found at the foot of a block of flats. Was it an accident, suicide or murder? Using the measured distance from the building together with elementary geometry and dynamics, the forensic scientist can form an opinion as to whether the deceased fell, jumped or was pushed.

Ballistics calculations, such as computing the ricochet angle of a bullet bouncing off a solid surface, use trigonometry. Bullet trajectories determine the distance from shooter to target, and perhaps the height of the gunman and where he was standing when he shot his victim.

#### Rates of change

The exponential and logarithmic functions, found throughout science, play a key role in forensics. The exponential function relates to processes that depend on the amount of material present as time changes. Rates of heating or cooling, or of the metabolising of alcohol and drugs, are governed by exponential rates of change.

After death, a body cools until it reaches the environmental temperature. By modelling the cooling rate mathematically, using Newton’s law of cooling, an exponential decay of temperature difference is found. This enables estimation of the time elapsed since death. In practice, more elaborate models can be used.

For quantities that vary over many orders of magnitude, such as the concentration of chemicals in the body, the logarithmic function allows us compress them into a more manageable range. The pH scale, which indicates the level of acidity, is of this sort and is often vital in forensic work.

Probability and statistics are of growing importance in law enforcement throughout the world. Quantitative statistical analysis is used to compare sets of experimental measurements to determine whether they are similar or distinct. This applies to glass fragments, drug samples, hairs and fibres, pollen grains and DNA sequences.

The chance of two people having identical DNA profiles may be one in 100 trillion, but forensic biologists must work with crime scene samples that are small and degraded. This makes the identification of a unique individual more difficult, and subtle probabilistic arguments are required to draw inferences.

When analysing evidence from fingerprints, blood groups and DNA profiles, probability enters the scene. Conditional probabilities are of vital importance in a forensic context. Such considerations may determine whether two events relating to a crime are linked or independent. The calculation of the probability of occurrence of multiple events is subtle if the events are related. Failure to understand conditional probability has led to some tragic miscarriages of justice.

In the hands of forensic scientists, every piece of mathematics we learn in school may prove to be a matter of life and death.

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