Measure three inches up from the floor, on the stringer side of the post. Mark the location in the center of the post.
Insert the 1-inch paddle bit in the drill. Drill a 1-inch-deep hole in the stringer side of the post at the mark you made.
Remove the 1-inch bit from the drill. Install the 3/16-inch drill bit.
Drill a 3/16-inch hole through the post and into the framing of the stair stringer. Drill this hole at an angle to penetrate the thick part of the framing. Center the point of the drill bit in the 1-inch hole, then adjust the drill approximately 15 degrees toward the stair framing as you drill the hole.
Replace the 3/16-inch bit with the 3/8-inch paddle bit. Drill a 3/8-inch hole through the post without penetrating the stair framing.
Place the washer on the lag screw. Insert the lag screw tip into the 3/8-inch hole.
Ratchet the screw in the hole by turning it clockwise until the washer seats tightly in the hole; the washer will slightly depress into the wood.
Cut a 3/8-inch piece from the 1-inch dowel. Apply wood glue to its edge.
Insert the dowel piece into the 1-inch hole. Sand the dowel flush with the surface of the post.
Check the post for plumb with the 24-inch level by placing it on perpendicular sides of the post.