A lamp has two bulbs, each of a type with an average lifetime of 6 hours

please see the attached file for the complete solution.

probability
________________________________________
A lamp has two bulbs, each of a type with an average lifetime of 6 hours. The probability density function for the lifetime of a bulb is f(t) =(1/6) e^(-t/6), t <= 0.

!!!! I think the pdf is

since this is an exponential distribution.

What is the probability that both of the bulbs will fail within 3 hours?

Solution. We can compute the probability one bulb will fail within 3 hours. It is

So, the probability that both of the bulbs will fail within 3 hours is

Note: We consider whether or not one bulb fails is independent of the failure of the other.