No, a hit by pitched ball is not an official at bat, just like a base on balls is not an official at bat. Yes. It will not show up underneath the AB statistic for a player but it does count towards the number of at bats needed to win a batting title. Actually, number of at bats is not the term used to determined a batting title. The term is actually "plate appearances" and includes official "at bats", walks, hit by pitched ball, catcher interference, sacrifices, sacrifice flys, which do not count as "at bats". Official "at bats" are recorded for a hitter who gets a base hit, reaches on an error, strikes out, or hits into an out. Batting averages are determined by dividing hits by official at bats, so, a hit by pitched ball does not count toward one's batting average as it is not an "at bat", but is a "plate appearance".
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No...a walk, either four balls pitched to a batter or the batter being hit by the pitch, does not count as an "at bat," and does not affect the player's batting average. It does, however, help his "on-base percentage."
Under MLB rules, at any point during an at-bat, a manager can instruct the pitcher to intentionally walk a batter -- even on an 0-2 count. Whether he SHOULD do so is another question, but doing so is not against the rules.
That is when you walk the batter on purpose, by intentionally throwing 4 pitches out of the strike zone, hopefully far out enough where he can't hit it.
No. An intentional walk is still marked in the books as a walk. They are plate appearances, but not at-bats.