I need to install a new wall receptacle which will be located on a interior wall in my bathroom. I need advice on drilling my chase hole. In my attic, there is a 2" x 4" turned on edge which i assume is the top plate for the wall in question. I will say that in past experiences the top plates I have worked with have been single or double 2"x 4" or 6's laid flat on top of the wall studs. Regardless, I will need to drill through this "plate" to run my new supply wire to the receptacle. I will also have to drill through several horizontal braces that run horizontally between the 2 wall studs where I will set the new box for the receptacle.I will have to use a flexible bit long enough to reach through the lowest brace to do this but would like to know if I will weaken the integrity of the top plate too much if I drill a 1/2" chase hole through it using a standard wood paddle bit since I will be drilling through the narrower width of the board.

Thanks for any replies.