Errors in Probabilistic Reasoning

 Conditional probabilities are of the form, if such and such is the case, then the probability of so and so is y. Stanovich reports an article described by Robyn Dawes that had a headline that said that marijuana use led to the use of hard drugs.1 The headline implied that the survey was the probability of a student using hard drugs given that the student had used marijuana. But the survey was about the inverse probability, the probability that the student had used marijuana given that he had used hard drugs. If you think about this for a minute, you will realize, if you haven’t already that they are not the same. Many experiment with marijuana, but few, fortunately, move on to hard drugs.

Unfortunately, the inversion of conditional probabilities is not restricted to the above article; it is a common occurrence. Both patients and medical practitioners sometime invert probabilities and think that the probability of a disease, given a particular symptom, is the same as the probability of the sympton, given the disease. Bear in mind that different diseases share many of the same symptoms, and keep this in mind when you are the patient.

Perhaps the most blatant use, or misuse, of inverse probabilities occurred during the O. J. Simpson murder trial. One of Simpson’s defense attorneys, Dershowitz, I believe, presented the conditional probability that a husband who had assaulted his significant other would eventually murder her. That probability, fortunately, is quite low (0.0004) . So this assertion buttressed the defense. The prosecution failed to recognize that it was the inverse probability, the probability that a husband who had murdered his significant other had previously assaulted her is very, very high (0.89). This presentation to the jury would certainly have buttressed the prosecution’s case.2

Please don’t mistake me for taking a position on the Simpson murder case. I hear the jury foreman, who happened to be a woman, state that had this been a civil case, where the standard is the preponderance of the evidence, they would have convicted. However, in a criminal case, the standard is beyond a “reasonable doubt.” Now I would appreciate a definition of a “reasonable doubt.” A jury risks making two errors: letting a guilty person go free and convicting an innocent person. So what is the standard here? Is mistakenly convicting one person in four beyond a “reasonable doubt.” One person in ten? One person in twenty? One person in a hundred? One person in a thousand? I think a definition is needed here to bring the legal system into the 21st century.

1Stanovich, K. E. (2009). What Intelligence Tests Miss: the psychology of rational thought. New Haven: The Yale University Press.

2Gigerenzer, G. (2002). Calculated Risks: How to Know When Numbers Deceive You. New York: Simon & Schuster

© Douglas Griffith and, 2009. Unauthorized use and/or duplication of this material without express and written permission from this blog’s author and/or owner is strictly prohibited. Excerpts and links may be used, provided that full and clear credit is given to Douglas Griffith and with appropriate and specific direction to the original content.


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