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Generally, LED's are very efficient, especially when they operate on either 9V or 12V power systems. There are LED lights that consume only 1 watt of power, and they SHOULD last for years, NOT months, particularly if you are only using them 4 hours a day. The diode itself does not degrade, it's usually the transistor (wiring) that fades out, eventually, and there are LED's that work off of rechargeable batteries, creating lots of light with very little power consumption.
The bulb has 2 terminals or connection points for wires , so does your switch and your battery. One terminal on each is for a live wire and the other is neutral. Take one wire from the left terminal of your battery and connect it to your left terminal on your switch , then wire from your switch to the left terminal of your bulb , run a wire from the left terminal of your bulb back to the battery to complete the circuit. Do the same with the right terminal in the same order and your done. Some battery terminals are top and bottom it doesn't matter which way you connect the wires as long as both wires start one end and finish the other.
MTBF can be calculated as the inverse of the failure rate for constant failure rate systems. For example: If a component has a failure rate of 10 failures per million hours, the MTBF would be the inverse of that failure rate. MTBF = (1,000,000 hours) / (10 failures) = 100,000 hours. Since LED's are "Solid State" devices and if they are used as prescribed by their data sheets they should last a very long time. Most devices fail due to excessive heat or mechanical vibration. Since LED's are packaged in a "Solid" housing and generate very little heat they usually last a very long time. But, that's not to say that if 1 million are in use that one might not fail after 1000 hours. It just means that "most" of them will not fail for a very long time. But, if you use one in a manner that exceeds the intrinsic properties of that device you will, no doubt, see a decline in the MTBF. Poor circuit design will be detrimental to any components MTBF. If you look at automotive brakes as an example. The MTBF of most automotive brakes is 35,000 miles. But, if you live in Phoenix AZ, and always drive on the "flat" freeways you may get 50,000 miles on your brakes. Where someone living in Denver CO, and drives in the mountains all the time may only get 15,000 miles.
