Rejecting an explanation for a particular event on the grounds that it requires a rare or unlikely event to have occurred, while ignoring that the favoured explanation might actually be even less likely. This fallacy ignores the fact that 'improbable' doesn't mean 'impossible'. Like the Gambler's Fallacy, this is also a statistical error.

As the name implies, this fallacy is a favorite of prosecutors in legal cases and sometimes in procedural shows like CSI -- it can be quite tempting to argue, "How likely is it that this really happened the way the defendant said it did, if the odds of it happening that way are 1 in 10 million? Which is more believable -- that he's lying or that something that improbable really happened?" It also lends itself well to CassandraTruth plots.

An argument of this form often ignores that unusual cases are, well, unusual. We tend to notice unusual events more than common events, and the very fact that the issue is being argued over guarantees that it is likely an unusual event. For instance, if a practised hunter accidently shoots his friend, one could argue that the odds of him making such a serious error is very small. But then, the alternative explanation is that the hunter ''purposefully'' shot his friend, which is also somewhat unlikely. In the end, the event itself can ''only'' be explained by one of several improbable explanations, and so the fact that they ''are'' improbable ceases to be relevant.

Wiki/{{Wikipedia}} has an article on the subject [[http://en.wikipedia.org/wiki/Prosecutor%27s_fallacy here]].

!!! Examples:

* A hypothetical example from Wiki/TheOtherWiki: if a DNA sample with a 1 in 100,000 chance of producing a match is run through a database of 1 million people, it will probably produce around 10 meaningless matches - on its own, it can't be taken as proof (the related defender's fallacy is to argue that this evidence should be dismissed for that reason).
* This was one of two errors in statistical reasoning that contributed to the result of the [[http://en.wikipedia.org/wiki/Sally_Clark Sally Clark]] trial in the UK. Sally Clark was arrested, charged, and wrongfully convicted of killing her two sons, who had died of sudden infant death syndrome, on the basis that two cot deaths in one family was extremely unlikely (an example of the prosecutor's fallacy -- double homicide isn't likely either). This error was compounded by an expert witness, who asserted that the probability of a double cot death was 1 in 73 million (a figure which assumed, without evidence, that both deaths were independent of each other -- ignoring possibilities such as a family with a genetic predisposition towards cot deaths).
* This was also a problem in [[https://en.wikipedia.org/wiki/People_v._Collins People v. Collins.]] A mixed-race couple (a black man with a mustache and beard and a white woman with blonde hair) were seen robbing an old woman and fleeing in a yellow car. The Collinses were a mixed race couple with the hair and car described. The prosecution famously claimed that the odds that such a couple existed in the area were 1 in 12 million, based on made-up statistics. [[note]]He asked his secretaries what they thought the odds were for a woman to be blonde, for a black man to have facial hair, and so on. He then gave these statistics to a mathematician, who treated each variable as if it was independent, even if it wasn't (men with beards are likely to have mustaches as well, but he treated them as independent variables).[[/note]] Even if the statistic was correct, and the likelihood of such a couple existing was 1 in 12 million, all it proves is that it is statistically unlikely for them to exist. Another couple, just as statistically unlikely, could have robbed the old woman. It doesn't mean that the chances that ''the Collinses weren't robbers'' was 1 in 12 million, though that was what the jury seemed to believe.
* Illustrated in creationist arguments. "The odds of everything happening just the way it has happened is infinitesimally small, so God must have created everything."
** This creates a FalseDichotomy. Either everything had to be exactly the way it is now, or there would be nothing at all. However, there is no reason to suspect that this universe is the 'jackpot', but rather that it is one of many possible outcomes, and it only has special value because it's the one we happened to have.
** And this is when their statistics are even valid, instead of recognizing that the naturalistic explanation is ''not'' due to random chance. (For instance, creationists will claim that the odds of a peptide chain folding into precisely the dimensions of a functional protein is absurdly low, completely ignoring [[http://en.wikipedia.org/wiki/Anfinsen%27s_dogma that it has been demonstrated]] that the natural state of proteins is the one that is thermodynamically most stable, and so will always fold that way.)
* This is also a favorite for conspiracy theorists when some (apparently) unlikely coincidence becomes part of the event in question. To use a UsefulNotes/WorldWarII example, one radar site picked up the Japanese aircraft headed toward Pearl Harbor and reported the contact but were dismissed because entirely coincidentally a flight of aircraft from mainland was due to arrive at roughly the same time. This has been used by conspiracy freaks to argue the Japanese were allowed to attack because the odds of that sort of coincidence seem so remote.
* The implicit assumption behind the ''Series/JudgeJudy''-ism "If it doesn't make sense, it isn't true."
* In ''The Poisoned Chocolates Case'' by Anthony Berkeley, Mr Bradley makes a list of twelve statements about the murderer, and declares that the odds against a random person meeting all the conditions are 4,790,000,516,458 to 1 against. But what he should be calculating is "What are the chances that, given that a particular person fulfills all the conditions, that person is the criminal?" -- which isn't the same thing at all. As Bradley goes on to point out that he himself meets all twelve conditions and is therefore, logically, the murderer, it's clear that he's only using the fallacy to {{troll}} his audience.
* In medicine, a test will have various numbers which indicate to the practitioner how much stock to put into the test's result. The four most commonly reported are: sensitivity (what percentage of people who have the tested condition test positive), specificity (what percentage without the condition test negatively), positive predictive value (what are the odds that a random positive answer means that someone is positive for the condition), and negative predictive value (odds a negative test means you don't have the condition). This fallacy is most similar to a situation where a test has a high sensitivity and high specificity, but a low positive predictive value.
** A concrete example: suppose an amazingly accurate test comes out that picks up on 99% of people with a disease and comes up negative in 99.9% of people without the disease. Now you test ten million people for a disease which occurs in 5 people per 100,000. There are 500 true cases of the disease. Your test identifies 495 of them. Your test also mislabels 1 in 1,000 as having the disease when they don't, which is approximately 10,000 people out of that 10 million! The positive predictive value is about 495/10,000, so a given positive test result only has an approximately 5% chance of actually identifying a person who has the condition.