A countable cup game and certifiable randomness

Consider the following game:

There is a countable sequence of face-down cups and an infinite bag of balls. First, player one places balls underneath finitely many of the cups. Next, player two turns over some (possibly infinite) set of cups and looks to see if they have balls underneath them. Finally, player two points to a cup that she has not yet turned over and declares “this cup does not have a ball underneath it.” Player two wins if her guess is correct; otherwise player one wins.

What’s a good strategy for player two in this game?

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