People vs Collins Solution Analysis
People v. Collins was a 1968 American robbery trial noted for its misuse of probability and as an example of the prosecutor’s fallacy. (Wikepedia, 2021)
Trial (Source: Wikipedia, 2021)
After a mathematics instructor testified about the multiplication rule for probability, though ignoring conditional probability, the prosecutor invited the jury to consider the probability that the accused (who fit a witness’s description of a black male with a beard and mustache and a Caucasian female with a blond ponytail, fleeing in a yellow car) were not the robbers, suggesting that they estimated the odds as:
Black man with beard 1 in 10
Man with mustache1 in 4
White woman with pony tail 1 in 10
White woman with blond hair 1 in 3
Yellow motor car 1 in 10
Interracial couple in car 1 in 1,000
The jury returned a guilty verdict.
Solution Analysis
The prosecutor made assumptions on the estimates for the individual probability of the characteristics that were not proven. Eg. Was the probability of an interracial couple in 1964 exactly 1/1000 or was there a possibility of it be higher or lesser? The necessary data and findings to support the estimates were not researched and presented to the court.
The prosecutor could have done a statistical sampling of a population to arrive at the estimated probabilities to better be able to connect the estimates to the real world. There seemed to have been no effort made by the prosecutor on this end. If the due diligence was done on data gathering to validate assumptions. Eg. the DMV records could have been checked to see the volume of yellow cars registered. In addition, observing the number of cars and people with the defined characteristics on different days and at different timings would have given an idea of the rarity or frequency of the couple’s characteristics. Taking this approach would have provided better probabilities that would have been estimated based on randomly drawn couples visiting the store or around the scene of the robbery. It would have helped to validate combinations of the characteristics.
The assumption of the individual characteristics being “statistically independent” was also wrong. In layman's terms what this means is that if A and B are two events then A event happening has no effect on the probability of B happening. A does not have any impact on the outcome of B. Eg. purchasing a car and then purchasing groceries. Relating to the case, it is more likely that a man with a beard has a mustache but a man with a mustache need not also have a beard. The product rule applied further gives a wrong calculation if people with a beard in that era also sported a mustache. So if the criteria of ¼ for men with a mustache is removed and combined with “black men with mustache and beard” then the product would have been 1/3,000,000.
Given the overall assumption on probability meeting, the criteria was 1/12,000,000. A better question to frame would be:
- Is there no other couple in LA in 1964 with the same characteristics (given the total population of LA in 1964 and accounting for any potential tourists)?
- What are the chances of more than one couple meeting the characteristics? A probability but does not ensure that the couple deemed guilty are the absolute ones who indeed are guilty.
Other possibilities include:
- The persons described by the persecutors could be wearing a mask.
- The woman who was seen by the old lady and that by the man could be 2 different persons.
- The calculation might further be incorrect based on the definition of a couple i.e. married or two individuals that happen to be together at that time.
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