The 3rd prong is a ground wire, which is ONLY actually used when there is an electrical fault in the appliance.
In addition, the adapter SHOULD have a tab to connect the ground prong to the screw in the outlet plate. If the box is grounded and/or has a metal conduit from the box to the panel, that can act as a ground.
Actually, if the box is grounded, it will cost less than $5 to install a grounded outlet in the box.
You can get SOME of the benefit of a grounded outlet by installing a GFCI outlet in the box. That should cost less than $20, and can protect 'downstream' outlets. Any such outlets should be marked "No equipment ground".