I recently noticed that many internet sites refer to fat chance and slim chance as interchangeable, and in some cases they might be, but the two don't have exactly the same meaning.
When someone says fat chance, they mean extremely little or no chance, but when someone uses slim chance, they mean there is some hope.
Example:
She has a fat chance of graduating.
(Her chances of graduating are very low to nonexistent.)
She has a slim chance of graduating.
(Her chances of graduating are low.)