Faking a pick-off move

[ChuckElias]

I hang out on the basketball side of this site, but I hope you'll take a minute to settle a matter of my curiosity. I am a big baseball fan, and I like to think I know the basic rules pretty well, but I can't figure this out.

I think that if a left-handed pitcher steps toward first base while on the rubber, he must throw the ball to first base or be guilty of a balk. Do I have that much right?

If that's true, why is legal for a right-handed pitcher to step toward 3rd base, fake the throw, but then turn around and throw to 1st base?

I've seen this maneuver used a lot (even tho it almost never works), but can't figure out why it's allowed.

I'm talking about MLB. Is the rule different at the HS or NCAA level?

Thanks,

Chuck