No outs. Next batter hits a ground ball to pitcher; pitcher throws to 2nd for force out, then short stop makes bad throw to 1st. Guy that originally started on 2nd scores on bad throw.
Is that earned or unearned?
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Two rules apply here. All rules are from the 2021 Rulebook.
First, is this an error? Yes, it is. Two rules apply; the rule that indicates it is an error to make a wild throw permitting a runner to advance beyond where they would have but for the throw, and an exception for double plays.
First, wild throw errors, 9.12a6:
(6) whose wild throw in attempting to prevent a runner’s advance permits that runner or any other runner to advance one or more bases beyond the base such runner would have reached had the throw not been wild;
Then the double-play exception is 9.13d3:
(d) The Official Scorer shall not charge an error against:
(3) any fielder who makes a wild throw in attempting to complete a double play or triple play, unless such wild throw enables any runner to advance beyond the base such runner would have reached had the throw not been wild;
Since the wild throw enabled a runner to advance beyond the base the runner would have reached had the throw not been wild, it is an error. (No error is charged based on the batter-runner reaching first base, however; that runner is considered safe on a fielder's choice. The error is solely based on the throw allowing the runner to score, unless the batter-runner took second on the overthrow.)
Second, is it an unearned or earned run? This is 9.16 (d):
(d) No run shall be earned when the scoring runner’s advance has been aided by an error, a passed ball or defensive interference or obstruction, if in the Official Scorer’s judgment the run would not have scored without the aid of such misplay.
So the question is, in the Official Scorer's judgement, take away the wild throw, but leave the batter-runner on first - so it is now 1st and 3rd, 1 out. What happened after that in the inning? If the next batter hit into a double play, ending the inning, then the run is unearned. However, if the next batter hit a fly ball to the warning track, the Official Scorer likely would have ruled that the runner on 3rd would've scored anyway, and so the run is earned. Same for a single (unless it was a bunt single, perhaps) or a home run.
Importantly, no outs changed as a result of the error - again, the batter-runner reaching first is not considered an error. So a sac fly would be possible in the alternate reality where the error did not occur. The benefit of the doubt goes to the pitcher here though - so if anything happens that may-or-may-not have allowed the run to score (like a bunt single), most likely the Official Scorer would consider the run unearned.
See the 9.16 intro for more on how the Official Scorer reconstructs the inning.