"Fat chance" is an English idiom that actually refers to someone having little to no chance at all of accomplishing something. It is often used as a phrase that stands alone as an exclamation following a sentence that describes the situation in question. This is an example of an ironic idiom, because although "fat" generally means large, someone who has a "fat chance" actually has very little chance. The phrase is used in exactly the same way as the idiomatic expression "slim chance," even though the two idioms would seem to completely contrast each other.
There are times when the English words spoken by someone are not to be taken at face value. This often occurs when people use idioms, which are short phrases whose accepted meanings often have little to no relation to the literal definitions of the words that comprise them. As a matter of fact, these idioms may even differ in meaning from the time when they were first originated. Their meanings come from the way they are used and understood in the culture. One such idiom that should certainly not be taken literally is the phrase "fat chance."
In actuality, someone who is attempting to do something and is described in this manner doesn't have much of a chance at all. There might be a bit of exaggeration involved, but the general gist of the phrase is that the person cannot possibly achieve what he or she has set out to do. For example, consider the sentence, "I've got a fat chance of finishing that puzzle with all of those pieces missing."
One common way for the phrase to be used is as an exclamatory sentence all by itself. When it is used in this way, it is often in conjunction with some preceding sentence or question that sets up the response. Using the phrase in this manner really emphasizes the futility of the situation for the person in question. As an example, someone might say, "Do you really think he will ask that woman out tonight considering how shy he is? Fat chance!"
This is one idiom where the literal definitions of the words would actually steer someone unfamiliar with the phrase away from its accepted meaning. Someone might think that having a "fat chance" actually means that the chance is large or robust. Instead, the phrase is a prime example of both irony and sarcasm, especially considering that it means the very same thing as the phrase "slim chance."