at random vs willy-nilly

at random and willy-nilly both are adverbs.

Word	Noun	Adjective	Verb	Adverb
at random	No	No	No	Yes
willy-nilly	No	No	No	Yes

As adverbs, at random and willy-nilly are synonyms defined as:

Other synonyms of at random include arbitrarily, every which way, haphazardly, indiscriminately, randomly.

at random (adverb)	willy-nilly (adverb)
in a random manner	in a random manner
	without having a choice