# How infrequent is “a non-zero chance”?

I misinterpreted the expression “a non-zero chance” as an emphatic way to stress that there was no possibility or likelihood of something happening.

• there is a non-zero chance that they will pay attention to some of the responses…

• there is a non-zero chance that the feedback may matter.

The author explained

NB: …the chance of them changing it is not zero and therefore must be more than that. I used a double-negative to emphasize that, while there is a chance, it probably isn't a good chance.

Me: It's confusing and ambiguous... why not just say "slim chance"?

NB: Because I prefer 'non-zero chance'; I like the phrasing of it. I like using double-negation as a way to minimize something. And for the record, thus far you appear to be the only person who has mentioned being confused by this phrase.

• Does “a non-zero chance” mean ‘more than zero’?

• Is the author right? Am I the only person who could misinterpret this phrasing?

• Is this phrasing peculiar to American speakers or do British speakers use this expression too?

P.S someone posted this link to an SE Mathematics question: Zero probability and impossibility but I didn't understand a thing

• non-zero - Having a positive or negative value; not equal to zero. ‘an extremely small but non-zero chance’ en.oxforddictionaries.com/definition/non-zero – user067531 Oct 28 '18 at 14:27
• I’ve always intuitively understood it to mean “a greater than zero chance”, and have only ever heard it used in those contexts. I think it would be hard for me to interpret it any other way, as probabilities can’t be negative, they can only be positive or nil, and the expression says it’s not nil. (But of course because we’re choosing to contrast it specifically with zero, instead of saying “a good chance” or “a decent chance”, then the implication is the chances are not zero, but not high either: slim). – Dan Bron Oct 28 '18 at 14:28
• Also, whoever you were talking to was misguided in his interpretation of the grammar. This is definitely not double negation. The zero isn’t negating, it’s quantifying, and the not is negating that specific quantification. It’s like saying “This house costs a million dollars. // Well, I don’t have a million bucks. // Ok, so how many bucks have you got? // Not a million, anyway” (only here, by contrast, the negation of a high number [in context] is implying it’s a low number). – Dan Bron Oct 28 '18 at 14:48
• @DanBron why don't you post an answer?? C'mon slow day here at home, the rest of the week I'm too busy to spend quality time on Q&As – Mari-Lou A Oct 28 '18 at 15:00
• @Mari-LouA, this way of speaking is not peculiar to Americans (or people from any other region), but it is peculiar to people whose education or profession makes them used to thinking of probability in quantitative terms. Such people may say that something has the probability of zero, when others would say that it is impossible. Once you get used to thinking and speaking in this way, it is natural to say 'non-zero chance' for something that is possible. – jsw29 Oct 28 '18 at 16:23

The miscommunication between the OP and her interlocutor is an example of what sometimes happens in interaction between people whose ways of speaking are shaped by an education or professional experience that revolves around analysing phenomena in quantitative terms, and those with other kinds of backgrounds. The differences that the latter group characterises as qualitative, may be quite spontaneously characterised as quantitative by the former. (So, no, the OP is far from being the only person to have had the experience of such miscommunication.)

Consider, for example, the difference between moving and standing still. To many people that is probably a very clear, definite, qualitative difference. Such people may speak of moving things as having this or that speed, but would never speak of the speed of an object that is standing still. A scientifically trained person, on the other hand, finds it quite natural to say that such an object has the speed that equals zero. In such a person’s conceptual framework, the difference between moving and standing still is merely quantitative; it’s the difference between having the speed of zero and having some other speed. A person who is accustomed to this way of thinking may feel compelled, when speaking of things that are in fact moving, to say that they have some non-zero speed; to people on the other side, ‘non-zero’ in such a context seems redundant, as they would never apply the concept of speed to motionless things.

Now, the same division can be seen when people speak of probabilities. The everyday framework for conceptualising them has the concepts such as impossible, possible (but improbable), probable (likely), certain. The differences among these at first appear to be qualitative, and are spoken of as such. People who are trained to analyse probabilities in quantitative terms, however, think of them as a continuum between zero and one. In that framework, something that is impossible has the probability of zero, something that is certain has the probability of one, and everything that is possible but not certain has some probability that is between these extremes. A person who is accustomed to that framework may feel the need to use the phrase ‘non-zero probability’ or ‘non-zero chance’ to make it clear that whatever is talked about is not impossible. To a person who is not accustomed to it, such a phrase seems strange, just like the non-zero speed in the above example. (Incidentally, to answer directly the question posed in the title, yes, non-zero in this context means more than zero as the scale of probabilities does not go below zero.)

So, saying that something has a non-zero chance is just a way of saying that it is possible, that comes naturally to people who have a certain educational or professional background, but may be confusing to those who don’t. Contrary to what the OP suspected, this way of speaking is not peculiar to Americans or to speakers of any other regional variation of the language. In fact, the whole matter is not specific to English, as analogous differences between people of different educational/professional backgrounds can probably be found among speakers of any language.

Now, the miscommunication between the OP and her interlocutor had another layer that was superimposed on this. The term non-zero chance, just like the more mundane term possible, by its meaning covers a wide range, from the probabilities that are just a sliver above zero, all the way to one. However, it would be strange and misleading (but not false) to use either of these terms if one knew that the probability is very high. Although these terms do not logically entail that the probability is low, they do imply it (in the loose, everyday sense of imply), or implicate it, or suggest it. In other words they convey the idea that the probability is low, as a matter of pragmatics, but not as a matter of semantics.

The author is "right" in the sense that in his/her books, zero chance is already a negative. There may be some merit in this, as witnessed by this dictionary entry for zero:

He said that his chances of getting the job were zero (= he had no chance).

-- source --

The word zero in this entry is an adjective. The meaning of non-zero should be intuitive then; or, as Lewis Carroll would have it, if one understands the term "zero chance", then one is bound to understand "non-zero chance", too.

There is some merit in using non-zero chance as opposed to slim chance or any other more precise qualification, especially if you really do not know what the chance might be. Non-zero chance commits to very little: merely, that you believe the event is possible.

The quoted entry might also be taken for supportive evidence for there being no essential difference in BrE and AmE usage.

Does “a non-zero chance” mean ‘more than zero’?

YES

non zero Oxford English Dictionary

‘an extremely small but non-zero chance

Is this phrasing peculiar to American speakers or do British speakers use this expression too?

I hear and use this In AmE frequently. My sense is to imply a minuscule chance, a slim chance, a small chance etc...

Here are some usages recently in AmE news/magazines:

“Here’s a pretty incredible fact: There is a non-zero chance that Donald Trump isn’t paying any taxes,” Clinton tweeted, just minutes after releasing her own returns. Time Aug 11, 2016

and

If he starts treating your mother in a way that suggests a potential for abuse, you should absolutely intervene, but there’s a non-zero chance that he’s just happily dating an older woman. Slate Jun 27, 2016

and

So it’s important to check as soon as possible with your local Social Security office to see if you do quality for a non-zero excess spousal benefit. Forbes Jan 2015

And here are BrE usages in the same sense I am supporting:

Economics, Uncertainty and European Football: Trends in Competitive Balance "For CB, each competitor in a match must really have a significant (i.e. non-zero) chance of winning."

and

A 2014 study of chess experts, including Carlsen, shows that they consistently attribute a modest – but non-zero – contribution of chance factors to game outcomes. The Guardian Sept 2016

In summary, it is used by both AmE and BrE speakers in the same sense, probably more predominately in the U.S.

• What are the different senses in AmEng and BrEng? – Mari-Lou A Oct 28 '18 at 16:33
• @Mari-LouA There aren’t any, so far as I know. I believe lbf put that in as a hedge to say “my comments may only apply to my own dialect, I am not sure they can be generalized”. But the fact that it inspires questions like yours is why I wish answers wouldn’t contain statements or assertions without corroborating evidence. If you don’t know something, leave it out. – Dan Bron Oct 28 '18 at 17:18
• attempted to address both comments. – lbf Oct 28 '18 at 19:04
• There is nothing in “non zero chance” that indicates a slim chance. It is maybe more likely that you would refer to something as non zero when it’s close to zero but 80% is also non-zero – Unrelated Oct 28 '18 at 19:15
• Note that nonzero doesn't necessarily mean strictly positive. When speaking about probability, sure, but it's used in other situations as well (e.g. a nonzero value). – JJJ Nov 5 '18 at 3:28

Probability can never be negative :

The probability of the outcome of an experiment is never negative, but quasiprobability distributions can be defined that allow a negative probability, or quasiprobability for some events. These distributions may apply to unobservable events or conditional probabilities.

Wikipedia

So a 'non-zero' probability simply expresses that there is a positive probability.

The author is mistaken that it conveys the idea of a 'small chance'.

• It is true that non-zero does not logically entail that the probability is small; it could be anything up to 100%. However, in virtue of standard conversational conventions it does imply (in the loose, everyday sense of imply), or implicate, or suggest, that it is small. In other words, it does not convey the idea of smallness as matter of semantics, but it does convey it as a matter of pragmatics. – jsw29 Oct 28 '18 at 16:09
• a 'small chance' is what is implied to this AmE reader. – lbf Oct 28 '18 at 16:22
• If it doesn't imply a "small chance" does it mean any number is plausible? If I say that there is a non-zero chance of something happening, am I saying its probability is equal or greater to one? – Mari-Lou A Oct 28 '18 at 16:39
• @Mari-LouA, probability of something cannot be greater than one (or 100%). Probabilities, if one thinks of them quantitatively, range from zero (i.e. impossibility) to one (i.e. certainty). Note that an event that has the probability of, say, 3%, is not at all probable. – jsw29 Oct 28 '18 at 17:49
• I think that in everyday speech non-zero chance really just means: there is some chance that. – Lambie Oct 28 '18 at 19:12

Literally, a non-zero chance means that something may happen. It is actually used in different ways, and you need to distinguish from context. And sometimes it's not entirely clear In your examples:

There is a non-zero chance that the feedback may matter.

There is a non-zero chance that they will pay attention to some of the responses.

The tone how it is said will likely tell you whether it is more sarcastically (yes, it's possible, but in my experience it's not going to happen) or just slightly hopeful (I don't think it will happen, but it might; let's hope for the best).

There is a non-zero chance that he is the villain we are looking for.

Unlike your examples, we don't expect someone to be a villain. We expect feedback to matter, or someone paying attention to responses. So here it is the other way round: We are saying that there is a reasonably good chance that he is indeed the villain.