I know that this is a common error in argumentation that people make, but I don't know if there is a term for it. It's when people argue from an event being remarkable because of its low probability, without considering the probability of having observed any event that's similar enough to the one they just observed to elicit the same reaction. Is there a name for this fallacy?

For example, a player plays a game in which they have three rolls of a six-sided die. They roll a "4" on a d6 for all three rolls. They remark, "wow! The chance of that happening is rare, therefore the probablitity must be the rare probability 1/216. As a deduction:

P1: I have rolled (4,4,4).
P2: Based on my emirical experience, this is rare!
C: Therefore, (4,4,4) occuring in 6^3 events indicates the rare probability 1/216.

In fact, they would have had a similar reaction if any of the digits 1-6 was rolled 3 times in a row, for which the chance is 1 in 36, not 1 in 216. In the extended circumstance, they may even wager on such a persuasive, but falliable reasoning in a game of chance.

The general fallacy is as follows: Suppose there's some class of events, E, of which the specific event e is a member, and all of which would elicit the same response from an observer. The probability of observing any event in E is p, while the probability of specifically observing event e, is q < p. An observer observes event e, and considers it a remarkable coincidence because of the low value of q, when in fact, they should be considering the value of p. Lastly they use a intuitionally fuzzy measure 'rare' which conflates q and p in drawing a conclusion.

  • Probably a language question. – puppetsock Feb 3 '20 at 19:50
  • It is not clear why they "should" consider a class of events and not a single event. What they should do depends on their goals, which we have no information on. 444 is statistically rare, so if they only care about 444 then its occurrence is "remarkable" to them. That other events they do not care about are also rare is moot. I suppose, they could be faulted for taking things out of context, but without knowing what the taking is for it is impossible to say what is or is not "in context". – Conifold Feb 4 '20 at 0:58
  • @Conifold I did put, as an assumption for the kind of situation I'm talking about, that they would have the same reaction for any event in the class of events. Indeed, if there was something special about "444", and not others, that would elicit a unique response then it's justified. But if they're merely responding to the fact that it's a triple, their 1/216 estimate is wrong. It may seem contrived to put, as an assumption, that they should care equally about all events in the class of events. But usually you know it when you see it. – Bridgeburners Feb 4 '20 at 14:07
  • I think something more cogent is needed to judge validity here than what they "care" about. Namely, their estimate is wrong for what? How are they supposedly planning to use it that would be in error? If the only "use" of it is getting excited I am not sure what "wrong" means exactly. I can think of people drawing conclusions based on mixing micro and macro state probabilities, for example, such as the common association of entropy with disorder. But I am not sure it is the same thing. – Conifold Feb 4 '20 at 21:44
  • This is not a logical fallacy, but retroactively picking a type of event to calculate the "improbability" of based on already knowing the outcome is considered a type of statistical fallacy, sometimes called the Texas sharpshooter fallacy. Physicist Richard Feynman used an informal example of this where he pretended to marvel over the unlikeliness of having seen a particular meaningless sequence of letters and numbers on a license plate. – Hypnosifl Oct 30 '20 at 23:46

Strictly speaking, an error is not necessarily a fallacy, it becomes one only when used to illegitimately support an argument. The observation your person is making isn't a logical fallacy --it isn't even factually wrong. You would have to make it the cornerstone of an argument for it to be a fallacy. What fallacy it would be would depend on how it was used.

For instance, supposed you would have to use the datum of rolling three "4s" in order to support a conclusion such as "I'm immensely lucky today." In that case, we might call this "cherry picking," which is using only the data that supports your conclusion.

  • In deference to your rating, could you clarify for me how it could be cherry picking in the case that gambler isn't aware that there is a difference in the probability of a single permutation and of a single instance of a class of permutations? – J D Oct 31 '20 at 3:04
  • I'm flattered by your deference to my high rep, but perhaps the more relevant stat here is the 0 upvotes this answer has earned in the last 10 months. With that said, the point is that a mistake is not a logical fallacy in itself. A logical fallacy is an illegitimate argument. The observation your person is making isn't a logical fallacy --it isn't even factually wrong. You would have to make it the cornerstone of an argument for it to be a fallacy. What fallacy it would be would depend on how it was used. – Chris Sunami supports Monica Oct 31 '20 at 3:24
  • Fair point, indeed, sir! :D Well met. – J D Oct 31 '20 at 3:29

Short Answer

I'm familiar with no precising definition for this persuasive, but erroneous line of thinking that could be used in an argument, however, arguably this is either equivocation of 'rare' or a conflation of the permutational and repetitional aspects of 'rare'.

Long Answer

This seems awfully close to cognitive biases that lead to the gambler's fallacy because we are dealing with a sequence of observations, and drawing a bad conclusion about the probability of the event. Let's instantiate your formalism for clarity:

A person rolls (4,4,4) and thinks, this is very rare, the probability is 1:3^6!

So we have two frames of reference because, first, it's perfectly true that the event has that probability, but, second, if "rare" is defined as any (n,n,n) were n is in the naturals 1 to 6, then there is a second concept at play. Let's examine two predications such that:

S1: (n1, n2, n3) where a single permutation is subsumed under 'rare'.
S2: (n,n,n) in the singular repetition of the element is subsumed under 'rare'

Is either statement counterintuitive or contradictory to general usage? No. For ease, let's define S1 as a 'permutational rarity', and S2 'repitional rarity' So, due to the inherent ambiguity of rare, we are calling both permutational AND repetitional rarity 'rare'. Now, what fallacy do we commit when used in support of an argumentation persuasively when we bungle a identical string in reference to two distinct concepts?

Well, if we were to say 'average' intending 'geometric mean' instead of it's arithmetic sibling, this clearly would be the fallacy of equivocation. But why shouldn't we consider this a conflation of probabilistic concepts, after all, we are confusing, two distinct definiens. As I can find no source that speaks to the nature of distinction between conflation and equivocation, I leave the OP to speculate.

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