Suppose I have a light-bulb whose risk of fizzling out and dying is constant over a time scale $[0,1]$. That is, suppose the time of light bulb death $\tau$ satisfies $$Pr(\tau\in[t,t+dt]\ \Big| \ \tau>t) = a\cdot dt,$$ where we have a conditional probability expressing the fact that there is zero probability that the light-bulb can die twice.
What is the expected time that the light-bulb will go out?
The condition $$Pr(\tau\in[t,t+dt]\ \Big| \ \tau>t) = a\cdot dt$$ means that $\tau$ has exponential distribution with parameter $a$.
The expected value for a an exponential distribution with parameter $\lambda$ is $\frac{1}{\lambda}$, so the answer is $\frac{1}{a}$.