Why the life of a light bulb follow an exponential law? Why the life of a light bulb follow an exponential law ? My teacher always said that, but he can't explain why it's this. So why did we decided that a light bulb follow an exponential law ? Where does this fact comes from ?
The exponential distribution comes from the assumption that light bulbs do not age. They just randomly, for each moment in time, decide whether to fail or to keep working, with the same probability regardless of how old they are. This leads to the exponential distribution.

Is this the way light bulbs actually works? I have no idea, but since this is such a ubiquitous example I would at the very least find it reasonable that it's close to the truth. Someone probably checked it once a long time ago, and it's just been the established truth since.
This was a great question, one I suffered with for a while.  If you accept the premise that a lightbulb can be modeled by a memoryless exponential function, then you are forced to accept some rather embarrassing results. 

For instance, let's say you have $2$ lightbulbs, each modeled by a memoryless exponential with a mean of $30$ minutes.  An experiment is done where the first bulb is run until it fails, the $2^{\mathrm{nd}}$ one is immediately put in and let run to it fails and you record only the time of the $2^{\mathrm{nd}}$ bulbs failure, let's say $60$ minutes.  Using Bayesian Inference, you want to use the time the  $2^{\mathrm{nd}}$ bulb failed to update you knowledge about when the $1^{\mathrm{st}}$ bulb failed. With Baye's Rule it is straight forward to show that you've learned nothing, that the failure of the $1^{\mathrm{st}}$ bulb is just as likely to have been at one second in, $59$ minutes in, or its mean, $30$ minutes in.  This is preposterous, and if you apply this to bulbs or even, worse bridges, you are making a big mistake.  

A coin flip is memoryless, the arrival of a single photon in a very low light environment is memoryless, not a lightbulb, because a lightbulb failure is not a single event, but rather a series of events where material is slowly lost from the filament finally leading to failure, and it cannot be modeled with a memoryless distribution.  I wish they would stop using lightbulbs as the cannonical example of exponential distributions and Bayesian Inference.