The antonym would be similar to those for "maximum" - minimum/minimal, least, smallest amount, least possible, lowest, nominal, token
She put only a nominal effort into completing the project, since it was very low priority.
They showed only a token interest in the artwork at the exhibition -- they were really only there for the free food and drink.
The new hire always does the least possible amount of work, just enough to avoid being fired.
I agree with the comments that "utmost chance" does not sound idiomatic. Usually "utmost" is used with some kind of effort, opinion, or trait, and not opportunity or outcome. Example:
I have the utmost faith that our client will complete the contract on schedule.
If talking about chance, opportunity, or expected outcome, I would instead choose a word like "highest" rather than "utmost". Antonyms of this would then be "least", or (as Mv Log says in comments) merest, million-to-one, minimal, outside, remote, slender, slight, slim.
She has only the slimmest chance of getting the job, but that doesn't stop her from hoping for the best.