Conditional Probability in the Courtroom
In his NYTimes Opinionator series titled Elements of Math, author Steven Strogatz writes about the often perplexing topic of conditional probability, which we know from class as the “probability that some event A happens, given the occurrence of another event B.” He notes that because this is such a “slippery” concept, it is often misused in the real-world. It’s easy for people to ask the wrong question or to calculate a probability that’s correct but leads to misguided conclusions.
In fact, during the 1994-1995 trial of O.J. Simpson, both the prosecution and defense fell into this trap. Each side asked the jury to consider the wrong conditional probability in their arguments. During the first 10 days of the trial, the prosecution presented evidence that O.J. had a history of domestic violence towards his ex-wife, which included battering her, throwing her against walls, and groping her in public, telling others passing by, “This belongs to me.” The prosecution
used this evidence to argue that spousal abuse “reflects a motive to kill” and is often the precursor to homicide. The defense, however, argued that the evidence of O.J.’s domestic violence was irrelevant because only a negligible fraction of men who abuse their partners go on to murder them (fewer than 1 in 2500 per year, or so they believed).
Both arguments were essentially asking the jury to focus on the probability that a man murdered his ex-wife, given that he had previously abused her. Was this the right number to look at? Not quite. This is because the purpose of the trial was to determine the probability that O.J. Simpson murdered his ex-wife, given two conditions: that he had previously abused her and that she was murdered by someone. This new probability, despite what the defense had suggested, turns out to be far from 1 in 2500.
To justify this, Strogatz uses the following example.
Imagine a sample of 100,000 domestically abused women. If we use the 1 in 2500 statistic estimated by the defense, then we would expect:
# of women to be murdered by their abusers in a single year:
(1/2500) * (100,000) = 40 women
# of women to be murdered by someone else:
given the US murder rate at the time of the trial was 1 in 20,000
(1/20000) * (100,000) = 5 women
In total, there were 45 female murder victims. Out of these victims, 40 were killed by their abuser. With some basic math it’s easy to see that the abuser was the murderer 90% of the time. Looks like that number isn’t as negligible as the defense thought.
This finding, however, wasn’t enough to convict O.J. of wrongdoing (ultimately the jury returned a verdict of not guilty), because the trial depended on a lot more evidence, such as claims that the police framed him or that O.J. and the killer had the same shoes, gloves, and DNA. Regardless, it’s interesting to see how, even in the most unexpected places, conditional probability has a way of showing up.
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http://opinionator.blogs.nytimes.com/2010/04/25/chances-are/