Tom
Agree with Jack. We used to put food coloring in the tank and see if the water in the bowl changed color.
Most newer ballcocks wait until the level in the tank drops to a certain point and then are fully open. This produces the sound you hear. If you take the top off the tank and rub the flapper (in the very center of the bottom of the tank and attached to the toilet handle) you will probably get a black residue on your finger.
As Jack said, replace the flapper.
