No it is no
The official scorer would have to determine if the throw was catchable. If the throw was, then the error would be on the first baseman and he would be charged as such. If on the other hand the throw was a bad one and the first baseman had to reach and could not catch the ball, then the error is on the pitcher. Only one error would be charged even though the base runner advanced two base and scored.
If he is called out on strikes and the catcher drops the ball, the batter may try to run to first base. In order to be put out, the catcher must throw the ball and get him out at first before he gets to the base. If the runner beats the throw, then the runner is safe. The pitcher is given credit for a strikeout. The catcher is given an error. This is how pitchers can have 4 strikeouts in an inning, which happened in MLB this week.
Error on the throw
yes
an error is only charged if the runner reaches base because of the mistake when they should have been out.
The runner is safe at third and the runner is safe at first.
Well, first of all, a dropped third strike is an out if the catcher's throw beats the runner. It's that way because that's just the rule.
This is a judgement call by official score keeper. If, in his opinion, the only reason the runner coming home from third was "safe" was because he out-ran the throw, then that would be considered an infield hit. It's the same situation as when a batter outruns the throw to first. If, however, the score keeper thinks the shortstop made a bad throw, or took to long to handle the ball, then it would be an error. In general, throwing to the "wrong" base is not an "official" error. Bad judgement, perhaps, but not an error.
No, the catcher (or any other player) can choose not to throw to a base at any time, regardless of whether a runner is going there. If the catcher knows he can't get a stealing runner out, he will most likely not throw to avoid the risk of an error.
you could check the runners at 2nd and 3rd, and then throw to first base. Once that is done, throw it home to prevent the runner on 3rd to go home.
This is umpire's discretion, but usually if the runner is in the runner's lane (the two parallel lines starting half-way up the foul line between home plate and first base) the runner will not be called out. However, if in the umpire's judgment the runner is in the runner's lane but is deliberately trying to interfere with the throw, the runner can be called out for interference.
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