The phrase "fat chance" can be used as a way of sarcastically describing the impossibility of something, but where did it originate from? I've googled it several times, and it always comes up with the definition, not the origin.
I believe this simply derives from one sense of the word fat:
fat 2a well filled out : of sizable proportions : THICK <a ~ letter> <a ~ volume of verse> : BIG <a resistor spark plug ... permits a wider gap, thus a fatter hotter spark — Newsweek> : unusually large <he had to pay a ~ price to move his factory — Martin Turnell> [Websters 3rd New Int'l Dictionary]
The term is used ironically. At face value it means there is a large chance of something happening, but underneath it really means there is a slim chance after all.
My English as second language wife jokingly asked about it the other day. My hypothesis (no research to validate) was that it could originate from probability theory. The chances (odds) of something happening are representing in statistics as a distribution. People are familiar with the shape of a normal distribution. https://en.wikipedia.org/wiki/Normal_distribution.
The more narrow the shape, provides a more narrow range and higher probability within this range of an occurrence. The fatter the shape distributes the odds of something happening across a much wider range with less chance for each # represented. Hence, a fat (wider) distribution represents a lower chance hence 'fat chance'