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Prosecutor's Fallacy
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  • One form of the fallacy results from misunderstanding Conditional Probability , or neglecting the prior odds of a defendant being guilty; i.e., the chance of an individual being guilty without specific evidence. When a Prosecutor has collected some Evidence (for instance a DNA match) and has an expert testify that the Probability of finding this evidence if the accused were innocent is tiny, the fallacy occurs if it is concluded that the probability of the accused being innocent must be comparably tiny. The probability of innocence would only be the same small value if the prior odds of guilt were exactly 1:1. In reality the probability of guilt would depend on other circumstances. If the person is already suspected for other reasons, then the probability of guilt would be very high, whereas if he is otherwise totally unconnected to the case, then we should consider a much lower prior probability of guilt, such as the overall rate of offenders in the populace for the crime in question, and the probability of guilt would be much lower.


  • Another form of the fallacy results from misunderstanding the idea of multiple testing, such as when evidence is compared against a large database. The size of the database elevates the likelihood of finding a match by pure chance alone; i.e., DNA evidence is soundest when a match is found after a single directed comparison because the existence of matches against a large database where the test sample is of poor quality (common for recovered evidence) is very likely by mere chance.


The terms "prosecutor's fallacy" and "defense attorney's fallacy" were originated by William C. Thompson and Edward Schumann in their classic article
[http://links.jstor.org/sici?sici=0147-7307%28198709%2911%3A3%3C167%3AIOSEIC%3E2.0.CO%3B2-9 ''Interpretation of Statistical Evidence in Criminal Trials: The Prosecutor's Fallacy and the Defense Attorney's Fallacy''].


EXAMPLES OF PROSECUTOR'S FALLACIES


Concrete examples are helpful to understanding the statistical reasoning behind these ideas:

1. Conditional Probability. Consider this case: you win the lottery jackpot. You are then charged with having cheated, for instance with having bribed lottery officials. At the trial, the prosecutor points out that winning the lottery without cheating is extremely unlikely, and that therefore your being innocent must be comparably unlikely. This reasoning is intuitively faulty — it could be applied to any lottery winner, even though we know ''somebody'' wins the lottery every day. The flaw in the logic is that the prosecutor has failed to take account of the low prior probability that you and not somebody else would win the lottery in the first place.

2. Multiple Testing In another scenario, assume a rape has been committed and that a sample is compared against 20,000 men who have their DNA on record in a database. A match is found, and at his trial, it is testified that the probability that two DNA profiles match by chance is only 1 in 10,000. This does ''not'' mean the probability that the suspect is innocent is 1 in 10,000. Since 20,000 men were tested, there were 20,000 opportunities to find a match by chance; the probability that there was at least one DNA match is

: 1 - \left(1- rac{1}{10000} ight)^{20000} \approx 86\%

which is considerably more than 1 in 10,000. (The probability that ''exactly'' one of the 20,000 men has a match is about 27%, which is still rather high.)


MATHEMATICAL ANALYSIS


We can view finding a person innocent or guilty in mathematical terms as a form of Binary Classification .

We start with a Thought Experiment . I have a big bowl with one thousand balls, some of them made of wood, some of them made of plastic. I know that 100% of the wooden balls are white, and only 1% of the plastic balls are white, the others being red. Now I pull a ball out at random, and observe that it is white. Given this information, how likely is it that the ball I pulled out is made of wood? Is it 99%? Not necessarily! Maybe the bowl contains only 10 wooden and 990 plastic balls. Without that information (the prior probability), we cannot make any statement. In this thought experiment, you should think of the wooden balls as "accused is guilty", the plastic balls as "accused is innocent", and the white balls as "the evidence is observed".

The fallacy can be analyzed using we see

  : <Math> Odds(IE) Odds(I)\cdot rac{P(EI)}{P(E\sim I)} </math>