Why is it an at bat if the batter hits gets thrown out at first but the run scores when it is not an at bat if the batter flys out and the run scores?

Answer 1

This is what you call and RBI. An RBI is short for "Runners Batter In"

Players compete for the highest amount of RBI's.

Since batting runners in help your team win the game, the at-bat does not count against the batter. That means that they just swipe the at-bat away. The batter still receives a Plate Appearance, but not an At-Bat. Plate Appearances do not effect your batting average or any of your averages, other then your on base percentage.

Think of it this way: If a player hits a fly ball and the runner tags up and scores a run, that's going to help his team win the Baseball game. Same goes for a ground ball out, as long as the run gets home, the batter is not charged with getting out. It will go on the score card as a RBI Sac Fly, or just an RBI for a ground ball.