The Prosecutor's Fallacy

The Prosecutor's Fallacy is a flaw in statistical reasoning that often arises in legal arguments, and the opposite of the Defense Attorney's Fallacy. The fallacy is to consider the unlikelihood of an event, while neglecting the consider the number of opportunities for that event to occur. Consider the following two examples:

  • You win the lottery jackpot. You are charged with bribing the lottery officials. The prosecutor points out that you are very unlikely to be innocent, since it's the probability of winning the lottery without cheating is so low. Obviously, the prosecutor's reasoning is faulty since it could be applied to any lottery winner.
  • In a murder investigation, a sample of blood found at the crime scene is compared against a DNA database of 20,000 men. A match is found, and this man becomes the primary suspect. The scientists running the test state that the DNA test will match by chance 1 time in 10,000. The prosecutor's fallacy is to assume that the suspect is innocent 1 time in 10,000. However, with 20,000 chances, the probability of finding at least one match by chance is:
    1-\left(1-\frac\{1\}\{10000\}\right)^\{20000\}\approx 86\%

What does this have to do with poker? The Prosecutor's Fallacy usually arrises when a player is losing. When opponents make a series of improbable draws, the player begins to grow paranoid and accuse the player of cheating, the game of being rigged, etc. However, if you play enough poker, it's likely that you'll have an evening where the cards never go your way. Put simply, if you observe a large number of events, it's likely that you'll observe some unlikely events!

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