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People vs. Collins

MATH, OF THE variety that you studied in school, and in the sophomore year of college, is today hot property in the fraud detection market. My friends in the CA world will vouch for this. But what many may not know is that its use dates back to 1964.  

Thanks to a stunning use of probability, the Jury found Malcolm Collins and his wife, Janet Collins, guilty of second-degree robbery. On appeal, the Supreme Court threw the conviction out of the window, while coming down heavily on improper use of math.

 

The case

Juanita Brooks, while returning home from shopping, was walking along an alley when she was pushed down by a person whom she couldn’t see. Soon she discovered that her purse, containing about $40, was missing. She filed a police complaint. A witness testified that the crime was committed by a two-person team, consisting of a man and a woman. That the man was black and that he wore a beard and a moustache. That the woman was Caucasian and that she wore a blonde hair tied in a ponytail. He also testified that they escaped in a yellow car. At the trial, things didn't go per plan for the prosecution. With everything falling apart, the prosecutors decided to use math to fix the accused. They were encouraged by the fact that historically the courts have looked upon math favourably as evidence.

 

Math model

The prosecutor intended to show that the probability of this two-person team of Mrs. and Mr. Collins fitting the description was very high. That they fitted the description and that they are hence guilty. Towards this, the prosecution invited a math professor, who said that the following probabilities were reasonable:

•    Black man with beard              1 in 10     i.e.    0.10

•    Man with moustache                1 in 4     i.e.    0.25

•    White woman with ponytail       1 in 10     i.e.    0.10

•    White woman with blonde hair   1 in 3     i.e.     0.33

•    Yellow motor car                      1 in 10     i.e.     0.10

•    Inter-racial couple in car           1 in 1000     i.e.    0.001

The professor told the jury what every student of Math 101 knows – that when events are independent, the probability of all of them happening is obtained by multiplying their respective probabilities. Therefore, in this case, the probability that Mrs. and Mr. Collins fit the description of the witness is:

= 1/10 x 1/4 x 1/10 x 1/3 x 1/10 x 1/100   

= 1/12,000,000 or one in 12 million.

The prosecutor argued that the chance that any couple would have the distinctive characteristics of the defendants was only one in 12 million, and that, hence, there is only one chance in 12 million that the defendants were innocent.

The jury returned a verdict of guilty.

 

The supreme court reverses …

The case went to the Supreme Court of California. Short of saying, “Bullshit,” the Court said that the professor’s technique was questionable on four counts. One, that there was no proof that the odds that he had given were right. Two, that there was no proof that the six factors named by the prosecution were statistically independent. Three, that the prosecutor had incorrectly framed the case and the evidence. And four, that there was no indication as to which one of the couples meeting this description was guilty of committing this robbery.

The court held that what needed to be proved is not the probability that the accused pair fits the description of the witnesses. The fact that they fit in was a fait accompli! What needed to be proved was what the probability that other couples would fit this description. And the court using complex math involving probability, population, etc., showed that while the probability that the accused fits the description may be 1 in 12 million, the probability of at least one other couple fitting the description might be as high as 40 per cent (computation is outside the ambit of this piece). Thus, the prosecution’s computations, far from establishing the guilt of Mrs. and Mr. Collins, showed that some other pair could be guilty!

It was not math that got thrown out of the window. It was the wrong and incomplete use of math that got thrown out of the window.

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