Is there a difference between "fat chance" and "slim chance," and if so, what is it?
Taken literally, they seem to be opposites, but do they mean the same thing?
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Both of these mean that something is very unlikely to happen.
That said, I believe the difference is more one of usage than of meaning.
Fat chance is an informal expression; it's the kind of language you might hear among friends. It can also function well as a standalone exclamation, sometimes carrying an undertone of amusement:
Do you think Donna would go out with me?
Slim chance has a nearly synonymous meaning, but a much more formal tone. It's the kind of language you'd expect to hear at a formal presentation:
Do you think your reseach shows that nanotechnologies might be hazardous to humans?
I believe there's only a slim chance of some undesired side effects that are not yet discovered.
What is the difference between "fat chance" and "slim chance".
These phrases don't mean the same. There is a difference in their meanings, though very small.
When you say he has a fat chance of getting this job, it means that he has an extremely little chance of getting this job (his chances of getting this job are one in a million). In other words, it means almost no chance. On the other hand, you use a slim chance to mean a slight/very little chance. The former phrase is indicative of almost no hope whereas the latter one is expressive of a ray of hope.
A slim chance means a very small chance. For example, suppose you have been playing chess for five years. Then you have a slim chance of winning the Australian chess championship this year. Calling the chance slim is a playful, metaphorical way of speaking, referring to a small probability as if it were a skinny (slim) person.
By analogy with slim chance, a fat chance means a very large chance. Only, notice the way I use it: Suppose you have been playing chess only for six months. Then you have a fat chance of winning the Australian chess championship this year!! It's sarcastic. The literal meaning is that you have a high probability of winning, but that's so absurd that it can only be understood as a joke. I'm calling it a high probability in order to point out how absurd it is to even think that you'd have any chance at all of winning. It's like saying "Nice car!" when you see a beater car.
People say slim chance as a playful metaphor. People only say fat chance as a sarcastic joke.
Notice that slim chance can be used to emphasize that the probability, though small, is greater than zero. But fat chance always emphasizes that the probability, though perhaps greater than zero, is very small.
(No offense intended to your chess-playing skills, and good luck in 2016.)
slim chance and fat chance are the same, although, contextually, they may not always be interchangeably used. Check this out: fat chance means small chance, also, slim chance means the same. While fat chance can be expressive of both irony and sarcasm, slim chance is purely literal. I believe it makes sense to verify the sameness and dissimilarity of words through their meanings and not by what they are expressive of.