risk vs run a risk

risk is a noun but run a risk is not a noun.

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

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

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

risk (noun)	run a risk (noun)
a venture undertaken without regard to possible loss or injury
the probability of being exposed to an infectious agent
the probability of becoming infected given that exposure to an infectious agent has occurred
a source of danger; a possibility of incurring loss or misfortune

risk (verb)	run a risk (verb)
expose to a chance of loss or damage	take a risk in the hope of a favorable outcome
take a risk in the hope of a favorable outcome