Probability 2 - Lightbulbs

Assume that a new light bulb will burn out after t hours, where t is chosen from [0,infinity) with an exponential density of f(t)= L*e^(-Lt)

In this context, L is often called the failure rate of the bulb.

a) assume that L = 0.01, and find the probability that the bulb will *not* burn out before T hours. This probability is often called the reliability of the bulb.

b) for what T is the reliability of the bulb = 1/2?

I set up part a as f(t) = 0.01e^(-0.01t), which is obvious, but I'm not sure where to go from here. Since this is the equation for the failure rate...would the probability of it NOT burning out be 0.99e^(-0.99t)? I am somewhat confused, my professor did not explain this too well.