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On your F150 the reverse lights are turned on by a range sensor on the left side of your transmission that tells the truck you've shifted into reverse. This sensor gets power from fuse 14 in the fuse box under the dash on the passenger side of your truck, the sensor turns on a relay in that fuse box that gets power from fuse 10 in that fuse box.
A wiring problem wouldn't be common, so I would start by checking both bulbs to make sure they are good, then check fuse 14, fuse 10, and the reverse relay in the fuse box inside your truck. I'll attach a picture of your fusebox with these fuses and the reverse lamp relay highlighted for you. The fusebox can be found behind the plastic panel that sits next to the passenger's right foot with sitting the the front passenger seat.
I hope I've been able to help you with your concern, if you have any other questions, or if you would like more information on your current question please feel free to ask.