The catcher fails to catch the third strike and it rolls away. The batter heads for first. The runner on 3rd advances toward home.
The catcher retrieves the ball, returns to home plate, and tags the runner out. The batter is safe at first.
Is this a passed ball, since the batter reached first on a strikeout, or is it a strikeout and a fielder’s choice, since presumably, the catcher could have chosen, poorly, to throw the batter out at first?
To be absolutely certain about whether or not it is a passed ball, you would have to see the play. If the catcher had no reasonable opportunity to catch the ball, (highly unlikely if it is a third strike), then it could be ruled a wild pitch, and not an error on the catcher. If, however, the catcher just boots it, and it should have been caught, then it would be a “strikeout” with an error on the catcher. Since the error occurred, it cannot be a fielder’s choice. The runner is safe at first on an error, and the runner is out at home on a failed attempt to steal.