run a risk vs take a chance

run a risk and take a chance both are verbs.

Word	Noun	Adjective	Verb	Adverb
run a risk	No	No	Yes	No
take a chance	No	No	Yes	No

As verbs, run a risk and take a chance are synonyms defined as:

run a risk and take a chance: take a risk in the hope of a favorable outcome

Other synonyms of run a risk include adventure, chance, gamble, hazard, risk, take chances.

run a risk (verb)	take a chance (verb)
take a risk in the hope of a favorable outcome	take a risk in the hope of a favorable outcome