There is always the possibility of bumping into someone in the supermarket but when we say "by chance" we mean accidentally, unplanned, without pre-arrangement. For me, none of the four choices really imply the same thing but of course you have to use one, and the word "luck" comes closest to "chance".
Poker is a game of chance. It is a question of luck whether or not you are dealt a good hand.
The lottery is a game of chance. If your numbers come up, you are lucky and if they don't, you're unlucky. Try your luck next time!
I was looking for a person to fix my car and, by chance, a flyer arrived through my door from a local mechanic. That was lucky!
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