lottery vs numbers game

lottery and numbers game both are nouns.

Word	Noun	Adjective	Verb	Adverb
lottery	Yes	No	No	No
numbers game	Yes	No	No	No

As nouns, numbers game is a hyponym of lottery; that is, numbers game is a word with a more specific, narrower meaning than lottery:

lottery: players buy (or are given) chances and prizes are distributed by casting lots
numbers game: an illegal daily lottery

Other hyponyms of lottery include tombola, lucky dip, numbers, numbers pool, numbers racket, raffle, sweepstakes.

lottery (noun)	numbers game (noun)
players buy (or are given) chances and prizes are distributed by casting lots	an illegal daily lottery
something that is regarded as a chance event