I think you are correct. The symptoms you described are that of a burnt out tail light. Remember, there are two elements in the bulb. One is a parking light the other for the signal and brake light. Have you tried replacing the bulb with a new one? Have you checked the light socket to see if there is power to it with the brakes?
That fast flashing front light indicates the rear light has an open curcuit which means the power isn't going through it fully. Check it with a test light or new bulb and let me know what you find.
Tuesday, April 16th, 2019 AT 12:06 PM