For some reason, that I don't understand, when I fill my bathtub my toilet starts to run. Why? What can I do to fix it?

What kind of pressure do you have in the house. Good pressure at all fuacets, or does the pressure drop dramatically when two or more fixtures are open?

You have a touchy toilet fill valve and the change in pressure when filling the tub is tripping the valve. It would cost $10 to replace or $0 to ignore

I thought that the fill valve was dependent on the ball float, and as such, couldn't be affected by pressure differentials. I suppose it wouldn't hurt to replace it anyhow.

Normally they do depend on the float, but yours is apparently malfunctioning and pressure difference is causing it to run. Pop a new one in and you should be all set. Let us know okay?