Here is a picture of the end on outside wall with notches.
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Both notches go past the wall support although I'm not sure they had to; I would have to take another look.As long as the notch does not extend to or past your support wall. The shear strength is what is the most important. That is the cross sectional area of your joist at the edge of your support wall. If the notch is past the support wall then your cross sectional area is the thickness of wood between the notch and the the support wall.
That again is no better than what the person that hacked this did. It is not the proper way of fixing.I would suggest going to a local home ceter and ask if they have anything like a truss plate. It is a perforated steel plate with sharp barbs, on one side that would be hammered into the wood. Possibly using a pair of these, one on either side of your joist, in conjunction with a polyurethane construction adhesive. If you push your electrical as far back as possible then place one truss plate on either side using a 2x6 or 2x8 piece of wood, sandwich the joist using a C-clamp to sink the metal barbs into the wood. I would spread the polyurethane construction adhesive on the barb side of the truss plate first. It would probably be a good idea to align the bottom of both truss plates with the bottom of your joist and try to center them on the edge of the horizontal 2x that your joist is on.
Bathroom on floor above is 5 feet wide. Joists run parallel to bathroom length. The joist to the left of these two is already doubled.He had a guy add 2 more 2x8s under the bathroom floor.
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Why just 2? You think that's going to help?
He just worked there. Before I left for work, I told him I thought the wire should be pulled and maybe a junction box added.That is too much taken out of the joist. You need to pull that wire and place a sister joist at that spot. To correct install wiring, only a hole needs to be bored, not a chunk that takes enough out to weaken the beam, which it is obvious that some has done that. Any other areas you can see that Bubba had his cousin did hack work on the joists?