Prosecutor's Fallacy

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 'statistically improbable' doesn't mean 'impossible'.

As the name implies, this fallacy is a favorite of prosecutors in legal cases -- it can be quite convincing 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?"^[1]

Examples of Prosecutor's Fallacy include:

• A hypothetical example from t'other wiki: 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).
• Occurred twice in a real life case from the UK: Sally Clark was accused of murdering her two children (both of whom had actually died of Sudden Infant Death Syndrome). An expert witness asserted that the probability of two cases of SIDS in one family was 1 in 73 million. Sally was convicted, but eventually cleared, although it took two appeals—the first, based on the incorrect 1 in 73 million figure, was quashed on the grounds that while the quoted figure was much worse than it should have been, it still illustrated that double cot death (SIDS) was very unlikely (with about 130 million births each year, one would expect one or two double SIDS death every year).
  • Note that this is also You Fail Statistics Forever, since the number was calculated by merely squaring the probability of one SIDS death, ignoring the possibility that they might possibly not be independent events.
• 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 is also a favorite for conspiracy theorists when some (apparently) unlikely coincidence becomes part of the event in question. To use a World War 2 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.
  • It also ignores a third possibility: that the Japanese knew of the aircraft movement in advance (entirely possible, as it was both scheduled and not classified) and timed their attack to take advantage of it. I don't know offhand if the Japanese actually did do that or not, but its certainly the sort of thing a competent planner would like to do.
  • There is still another explanation. Negotiations were getting near the brink and everyone could see there was a chance of a big blow up in the Pacific at the time. Naturally a lot of military traffic would be moving about. Much as if Kasparov moves his King Pawn next to Deep Blue's King Pawn and Deep Blue gets an opening that does not prove Kasparov is throwing the game but only that both are playing chess.

1. ↑ You can see why the fallacy is used liberally in Cassandra Truth plots.