Taking a chance with chances
Photo by dylan nolte on Unsplash
By Rien Kleijnen
Imagine a candy store that sells 100 different pieces of candy. One day, ten customers visit the store each buying one piece of candy at random. What is the chance that two of these ten customers end up buying the same piece of candy? Now imagine this candy store has a total of 24 employees. What is the chance that two of these employees have their birthday on the exact same day? Think about these questions for a moment.
At first glance, many people consider the chance of such incidents happening is rather low, and perhaps you feel the same way. However, the answer to these questions might surprise you: there is a 37.2% chance of two of these ten customers buying the same piece of candy, and around a 50% chance of two of these 24 employees sharing their birthday. At this point, you might find yourself puzzled wondering: how could your intuition about these questions be so off? The reason is that human reasoning is easy to fall prey to statistical misinterpretations like these. In this case, people tend to estimate the chances being low because the number of total candies in the store (100) exceeds the number of customers (10) with quite a large difference. However, this is not a reason to assume it is, therefore, unlikely for two of these ten customers to end up with the same piece of candy. The same goes for the second question, where the number of days in a year (365) is considerably larger than the number of employees (24). Unfortunately, statistical misinterpretations like these are also common in forensic evidence analysis and can lead to dramatic consequences for a potential innocent suspect. One such case is that of Ray Krone that took place in 1991.
It was almost New Year’s Eve in Phoenix, Arizona when the body of a 36-year-old woman was found. She was found dead in the men’s restroom of the bar where she worked. There was little physical evidence of the perpetrator. The only traces investigators could find bite marks on werre the neck and breast of the victim. They, therefore, turned to the friends and family for clues. After consulting with some friends of the victim, it did not take long before investigators were pointed towards Ray Krone, who allegedly helped the woman to close the bar in the evening before the murder. Krone, who has now become the prime suspect, was asked to cooperate and create a bite mark sample on a piece of Styrofoam. After investigators compared Krone’s bitemark with those found on the body, a sufficient degree of similarity between the two marks was established: the bite marks matched. The investigators stopped looking for suspects and Krone was arrested and charged with murder, kidnapping, and sexual assault. A year later, while maintaining innocence, Krone was sentenced to death for the conviction of First-Degree Murder and Kidnapping. This may look like a fair and reliable approach to identifying the perpetrator of a crime like this one. However, as you will see this case is a clear example where a statistical misinterpretation had adverse consequences for an innocent suspect.
The investigators’ line of reasoning was based on the assumption that for the bite marks on the body and the ones produced by Krone on the Styrofoam to match, they must come from the same source. This is called the assumption of individualization. However, this assumption leads to fallacious thinking because of the problem of induction, which is a common problem in forensic evidence analysis. Induction refers to a process of reasoning where conclusions are drawn about the whole based on a sample of observations. For example, if the only swans ever spotted were all white, one could conclude all swans are white. The problem here lies in the fact that we can never be sure that all swans are white as long as we have not seen all swans. Similarly, making the inference that the bite marks on the victim and Krone’s Styrofoam teeth impression share the same source can only be done if one has tested and knows for sure that every human produces a unique bite mark. That means, investigators would have to match the bite marks of the body with a Styrofoam impression of teeth from all the people in the world. Only then, by going around the problem of induction, uniqueness could be confirmed with full confidence, even though this is an impossible job. The question, however, is not necessarily whether all bite marks are unique, but if the bite mark analysis methods and procedures used are able to detect uniqueness. If all bite marks are in fact unique, it does not prevent an imperfect forensic analysis method like bite mark analysis to ascribe a high degree of similarity between several unique bite marks. Two impartial bite marks for example that look the same could mistakenly be identified as a match. This categorical approach of identifying suspects could be helpful to exclude potential suspects but could cause an innocent suspect to end up behind bars based on blindly assuming uniqueness of traces and imperfect uniqueness detection by forensic evidence gathering approaches like bite mark analysis.
Indeed, Krone became a victim of the individualization fallacy. After serving ten years in prison Krone got exonerated when DNA testing proved his innocence. The DNA source matched a man named Kenneth Phillips, who was at this point already incarcerated for an unrelated sexual crime. He lived close to the bar at the time the body was found. Just because it seems unlikely that, despite the large number of possible bite marks, two matching traces are merely a result of coincidence, does not mean that uniqueness can simply be assumed. Just because a store sells 100 different pieces of candy does not mean it is unlikely that two out of ten customers end up with the same piece of candy.
References
Brysbaert, M. & Rastle, K. (2013). Historical and conceptual issues in psychology (Second edition). Harlow, UK: Pearson.
Broeders, A. P. A. (2006). Of earprints, fingerprints, scent dogs, cot deaths and cognitive contamination - a brief look at the present state of play in the forensic arena. Forensic Science International, 159(2-3), 148-157.
Diaconis, P., & Mosteller, F. (1989). Methods for Studying Coincidences. Journal of the American Statistical Association, 84(408), 853-861.
https://innocenceproject.org/cases/ray-krone/
Saks, M. J., & Koehler, J. J. (2008). The individualization fallacy in forensic science evidence. Vanderbilt Law Review, 61(1), 199-220.