@MickeyB et. al.
Comment 1: Spring for a < $20 receptacle tester to find out what you really have at each outlet before you do much of anything else. It should have 3 lights to tell you if the outlet is wired correctly and also if there is a GFCI anywhere in the circuit between that outlet and the rest of the world. I use an Ideal #61-501 but other options are available.
Comment 2: A great companion to the above tester is a signal injector so you can trace the circuit back to the panel. (Also use to check for other outlets on the same circuit, so you can locate a second circuit in a given space, or find other loads that might affect the outlet you are investigating. - e.g., a bedroom window A/C unit that throws a crazy spike onto the line when starting/stopping.) WallyWorld and others offer a Klein Tools M2O41519KIT Digital Circuit Breaker Finder and Accessory Kit for about $80 which includes a version of the Receptacle tester mentioned above, a signal injector so you can find which fuse or breaker feeds the outlet in question, and some accessories so you can easily check light sockets, 2-prong outlets as well as 3-prong, and alligator clips to check for hot wires anywhere in-between or before you start putting things back together. Just don't buy anything without the correct approval label for your country! UL for USA; CSA for Canada, etc.
Comment 3: Before proceeding with poking around inside a wall receptacle box, turn the power OFF. Even some quite old houses that still have 2-hole outlets (no ground prong) actually may have a ground available at the receptacle box! BX, or as currently known MC or AC (armored cable), has been used widely in the US since the 1920's. You can look for any armored cable if you are lucky by using a flashlight to peek around the receptacle and look for a metal spiral-wrap sheath coming through a clamp at the back of the box with the wires to the outlet inside of the metal sheath. Better yet, CAREFULLY remove the outlet itself so you can actually see what wires go where. Old wires may be very stiff, and the insulation may crack.
I happen to live in a building where there are properly grounded 3-hole outlets, but only two wires coming to the actual receptacle! The ground is provided by the spiral- wrap cable sheath which is clamped to the box, and the ground point in the outlet is connected when the mounting strap of the outlet is screwed to the box.
You may have come across old two-prong adapters that will go into a two-hole outlet, and have a pigtail wire or a very short metal tab that are meant to be connected by being tightly clamped under the screw in the center of the outlet thus providing the ground path for the ground prong hole in the adapter.
Always check for a good ground in this type of a circuit because over the years the ground connections may have been compromised by oxidation, thermal expansion / contraction or even building movement due to traffic, settling, etc.
Comment 4: If you have ANY doubts, contact a Licensed Electrician in your area. I worked under a Master Electrician (who was NOT my dad) for many years and I respect electricity. My (extremely sweaty at the time) dad almost died when hit with 4000V and was VERY lucky he could leave the hospital a month later. YMMV.
