Question about Hep B Test Results

I have had three Hep B vaccine shots.  2 months after the last
vaccine shot, my lab results were as follows:

Hep B Surface Antigen  NON-REACTIVE
Hep B Core IGM Antibody NON-REACTIVE

4 months after the last shot, my lab results were as follows:

Hep B Surface Antigen  NEGATIVE
Hep B Core Ab, Tot     POSITIVE
Hep B Surface Ab       > 150 mIU/mL

Does it mean that I am immuned to the Hep B virus?  

Does the negative Hep B Surface Antigen result mean that I
do not have an active infection?

What test do I need to take to make sure that I am not a
carrier of the Hep B virus?