lottery vs game of chance

lottery and game of chance both are nouns.

Word	Noun	Adjective	Verb	Adverb
lottery	Yes	No	No	No
game of chance	Yes	No	No	No

As nouns, game of chance is a hypernym of lottery; that is, game of chance is a word with a broader meaning than lottery:

lottery: players buy (or are given) chances and prizes are distributed by casting lots
game of chance: a game that involves gambling

Other hypernyms of lottery include gambling game.

lottery (noun)	game of chance (noun)
players buy (or are given) chances and prizes are distributed by casting lots	a game that involves gambling
something that is regarded as a chance event