﻿ Mathematics and Probability

Mathematics and probability
E-mailed to Portia by Marco de Innocentis

"Innumeracy - mathematical illiteracy and its consequences" by John Allen Paulos, Professor of Mathematics and Presidential Scholar at Temple University in Philadelphia.

A tendency to drastically underestimate the frequency of coincidences is a prime characteristic of innumerates, who generally accord great significance to correspondences of all sorts while attributing too little significance to quite conclusive but less flashy statistical evidence. [...]

Coincidences and the Law
In 1964 in Los Angeles a blond woman with a ponytail snatched a purse from another woman. The thief fled on foot but was later spotted entering a yellow car driven by a black man with a beard and a moustache. Police investigation eventually discovered a blond woman with a ponytail who regularly associated with a bearded and mustachiod black man who owned a yellow car. There wasn't any hard evidence linking the couple to the crime, or any witnesses able to to identify either party. There was, however, agreement on the above facts.

The prosecutor argued that the probability was so low that such a couple existed that the police investigation must have turned up the actual culprits. He assigned the following probabilities to the characteristics in question: yellow car - 1/10; man with a moustache - 1/4; woman with a ponytail - 1/10; woman with blond hair - 1/3; black man with a beard - 1/10; interracial couple in a car - 1/1,000. The prosecutor further argued that the characteristics were independent, so that the probability that a randomly selected couple would have all of them would be 1/10 x 1/4 x 1/10 x 1/3 x 1/10 x 1/1,000 = 1/12,000,000, a number so low the couple must be guilty.

The jury convicted them.

The case was appealed to the California supreme court, where it was overturned on the basis of another probability argument. The defence attorney in that trial argued that 1/12,000,000 was not the relevant probability. In a city the size of Los Angeles, with maybe 2,000,000 couples, the probability was not that small, he maintained, that there existed more than one couple with that particular list of characteristics, given that there was at least one such couple - the convicted couple. On the basis of the binomial probability distribution and the 1/12,000,000 figure, this probability can be determined to be about 8 percent - small, but certainly allowing for reasonable doubt. The California supreme court agreed and overturned the earlier guilty verdict.

Whatever the problems of the one in 12,000,000 figure, rarity by itself shouldn't necessarily be evidence of anything. When one is dealt a bridge hand of thirteen cards, the probability of being dealt that particular hand is less than one in 600 billion. Still, it would be absurd for someone to be dealt a hand, examine it carefully, calculate that the probability of getting it is less than one in 600 billion, and then conclude that he must not have been dealt that very hand because it is so very improbable.

(from Chapter 2: "Probability and Coincidence")

Now - who was it that said the law is an ass? If it is possible or probable you did it, a jury WILL convict.

Posted June 2000

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