Does "take a chance" have the same meaning as "take one's chances" as they both mean to "gamble"?--
Roughly. 'Take a chance' adheres more to the concept, I think.
One entry of "take a chance" in my dictionary: "I'll take a chance with the weather and go for a pinic." Why is that "with" is used, and not "on."-- OK, it appears to be another possible preposition, but 'with' generally applies when its object contributes to the chance: 'I'll take a chance with my old clunker, and hope that it gets me to St. Louis.'
Just on a curious hunch, does "at" as in "take a chance at" show a persisting action in reaching your goal? Or it's an alternating use of at/on, where there's no significant difference between them?-- I see no such difference.