Mathematics and the Real World by Zvi Artstein

Author:Zvi Artstein
Language: eng
Format: mobi, epub
Publisher: Prometheus Books
Published: 2014-07-31T22:00:00+00:00

The fact that a donor's blood sample is tested, in other words the subject, that is, the donor, is chosen almost randomly, is an important fact of the analysis (it pertains to the assumption we made when introducing the formula in the previous section). If the subject had been sent for the test because he was suspected of being an HIV carrier, for example because he showed certain symptoms, the probability that he was really a carrier would be different; to find it we should use the original Bayes's scheme.

Why do most people asked the question consider that the chances that someone who gets a positive result in the test (as a carrier) really is a carrier is 99.75 percent? The reason lies in the way the brain analyzes the situation, a way that is inconsistent with the mathematical logical understanding of it. The brain perceives certain data and decides intuitively which are important, without undertaking an orderly analysis of the information. It does not look for missing information. Evolution instilled in us, or more precisely instilled in our subconscious minds, the recognition that it is generally not worthwhile to devote the effort necessary for a rigorous analysis of the problem. Therefore the brain concentrates on one prominent piece of information: the chance of an error is only a quarter of one percent.

This type of error is not confined to medical tests. Courts tend to convict someone who confesses to murder even without corroborating evidence. The reason that judges give is that the probability that someone will confess to a murder he has not committed is negligible. That fact is correct, but the statistical conclusion is not. To illustrate: assume that only one person in one hundred thousand will confess to a murder he has not committed (and taking into account the conditions of police interrogation that suspects undergo, that assumption is certainly not an overestimate). Assume also that someone is arrested randomly from a population of four hundred thousand, and he confesses to a murder committed the previous day. The chance that he is the real murderer is only 20 percent! The chance that the real murderer was found is only one in five (the murderer himself, if he confesses, and the other four of the population who would confess even if they have not committed the murder). With a larger population, the chances are even lower. The mistake the judges make lies in the fact that they only examine the chances that someone has been arrested who has a tendency to admit to a crime he did not commit; but once the suspect confesses, that probability has no significance. Once a suspect has confessed, what matters is how to differentiate between those who would make a false confession and the real murderer. That can be done by means of additional evidence or suspicious circumstances. Judges often overlook this distinction. In an article published in the Israeli journal Law Studies (Mechkarey Mishpat) in 2010, Mordechai Halpert and Boaz Sangero

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