Intuitively, it should just be $\frac{E[A]}{E[A] + E[B] + E[C]}$ as in the long-run, since the probabilities of picking the bulb are symmetric, each bulb will be used the same number of times. How do I formalize this using renewal-reward theory?
I would like to define a renewal as when a bulb burns out. Then a cycle is the time between $2$ bulbs burning out. If I can find the expected length of a cycle, then I am done.
If we have bulb $A$ right before a renewal, then with probability $1/2$ bulb $B$ is chosen with a lifetime of $E[B]$ and with probability $1/2$ bulb $C$ is chosen with a lifetime of $E[C]$. Thus the expected length of the cycle, given that we start with bulb $A$, is $\frac{E[B] + E[C]}{2}$. Similar reasoning works if we start with bulbs $B$ or $C$. But how do I use this to find the expected cycle length?