Halogen street lamp fixtures out of commission? We have 2 'eye' halogen desk light fixtures in our bathroom that...

We have 2 'eye' halogen desk light fixtures in our bathroom that own quit working. (Little halogen bulbs with the 2 prongs) We tested the bulbs within other fixtures- and they're fine. When you flip the switch you can hear a hum- so there is power going to the fixtures. What might be wrong beside them? Should I be able to fix/ replace parts myself or do we call for an electrician? Thanks!
Answers:    You should know from buying the things whether your fixtures are low voltage (with a modest sized box connected) or high voltage (direct wire) (Two prong bulbs come both ways.)
If low voltage, the transformer/power supply box have probably gone bad and you should know how to buy and replace it using normal not dangerous wiring practices. Find it, run it out, match it at the store.
If the power is direct connect, afterwards it is probably the socket and the moisture in the hip bath that is cause problems and you will have to replace at lowest possible the socket - and not all electrical stores convey them.
I think the oil lamp fixture is probably bad. I enjoy a light lately like that and it does NOT hum. Humming wishy-washy fixtures are not a good point.