Say I have a light bulb that can be on (A) or off (B). It alternates between being state A or B. It will be in state A for a duration a ~ exp(α), and in state B for duration b ~ exp(β), (parameterized by mean), back and forth

I have a hunch that P(state at time t = A) approaches α / (α + β) as t -> infinity.

**How can I find the exact probability of being in state A (or B) at time t, given the state at t0?**. There are uncountably infinite possible sequences of state durations between now and time t

and what general type of problem is this? poisson something? markov something?

Thanks