Information AboutProsecutor's Fallacy |
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The prosecutor's fallacy is a Fallacy of statistical reasoning that takes several forms.
The terms "prosecutor's fallacy" and "defense attorney's fallacy" were originated by William C. Thompson {Link without Title} and Edward Schumann in their classic article "Interpretation of statistical evidence in criminal trials: The prosecutor's fallacy and the defense attorney's fallacy." Law and Human Behavior, 1987, 11, 167-187. WHY THIS IS FALLACIOUS: SEVERAL EXAMPLES 1. A concrete example may make it clear why this reasoning is fallacious. Suppose there is a one-in-a-million chance of a match given that the accused is innocent. The prosecutor says this means there is only a one-in-a-million chance of innocence. But if everyone in a community of 10 million people is tested, one expects 10 matches even if everyone is innocent. If no other evidence is available (i.e. the accused has been charged solely on the basis of this match) then this match represents only very weak evidence of their guilt. 2. In another scenario, assume a rape has been committed and that a sample is compared against 20,000 men that 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 : 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.) Another perspective on this example is that it presumes the guilty party's DNA is in the database-- any of the other 0.01% of the population of men matching the profile might also be guilty. This notion matches the thought experiment described in the first paragraph of the next section. 3. Now 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 clearly faulty: the prosecutor failed to mention that cheating lottery winners are much more rare than honest winners. 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 actually white. Given this information, how likely is it that the ball I pulled out is made of wood? Is it 99%? No! Maybe the bowl contains only 10 wooden and 990 plastic balls. Without that information (the ''a priori'' 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 | ||
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