Bayes’ Rule and the Law
As we learned in class, Bayes’ Rule is all about calculating probabilities of events, given some that some other events have occurred. This simple formula has far reaching applications in fields such as medicine, sports, gambling, and politics. Many of these topics have been discussed in previous blog posts, or in lecture.
In his book “The Drunkard’s Walk,” Leonard Mlodinow discusses how Bayes’ Rule has been used (and misused) in a few different court cases. One case is about Sally Clark, a mother who had two children died of SIDS, one at 11 weeks and one at 8 eights. Law enforcement found it suspicious that she would have two children both die of SIDS, and so she was arrested and charged with smothering both children. The prosecution’s argument was that the probability of a child dying from SIDS is 1/8543. Thus, the probability of two children dying from it must be (1/8543)^2, or about 1 in 73 million. They reasoned that this meant Clark had only a 1 in 73 million chance of being innocent. On this argument alone, Clark was convicted. It wasn’t until two years later that the Royal Statistical Society found the case, and realized what a mistake the prosecutors had made. First, they had assumed that the events were independent. That is, they assumed that the first child dying of SIDS did not affect the probabilities of the second child, ignoring any genetic or environmental causes SIDS may have. Using Bayes’ formula would have fixed this, because they would have had to calculated the probability of the second death given that the first death had already occurred. More importantly, they never calculated the probability associated with a mother smothering two children. A representative from the Royal Statistical Society concluded that a pair of infants are 9 times more likely to die from SIDS than from murder, giving Clark only about a 1/10 chance of being guilty (assuming no further evidence is found). With this revelation, Clark appealed the case, and eventually was released from jail. (As it turns out, the prosecution had hid evidence that the second child had been suffering from a bacterial infection, which of course would raise the probability of SIDS).
This isn’t the only trial included in Mlodinow’s book. He includes two more examples where prosecutions and defense teams used similar probabilistic arguments to further their cases. In each trial, the jury believed the faulty mathematics, and reached a likely incorrect verdict. These cases should serve as testimony to the fact that educated and informed citizens are necessary to maintain a safe and healthy country. Had a single member of the jury in any of these cases known about Bayes’ Rule, they could have changed the outcome of the trial. Education doesn’t just helped the educated; it helps everyone.
Source:
Chapter 6 of “The Drunkard’s Walk: How Randomness Rules Our Lives” by Leonard Mlodinow.
https://books.google.com/books/about/The_Drunkard_s_Walk.html?id=UJxRLCq9l3IC
nice post!