Does the anti-roll bar reduce the chances for a car to tip over?

Not really, if by tip over you mean roll over onto its side.  The anti roll bar connects the wheels on opposite sides and increases the force on the more heavily loaded outside wheel in a turn. The body will roll less, but this is desirable because it keeps the tires in better contact with the ground. In a tall car more lateral grip means the car may roll over rather than slide, which is bad.

If you were thinking of a roll bar rather than an anti-roll bar, the answer is also no. A roll bar is meant to prevent the roof from collapsing if the car does roll over.