A bulb at the entrance of a building has a lifetime of exponential expectancy $100$ days. When burned, the concierge immediately replaced with a new one. In addition, an employee who takes care of the daily maintenance rempalce the bulb with a new one as a precaution according to a Poisson process with intensity $0.02$ per day. Determine : (a) the rate of replacement of the bulb? (b) the probability that the next bulb is replaced by the concierge
I succeed to do (a) and I obtain $0,01 + 0,02 = 0,03$. However, I blocked on (b) since a good while.
I'm not very aged ($13$ years old) and often I need a little help to solve a problem. This question seems difficult to do and I do not know how to approach it. Does anyone could provide a detailed and complete answer to this question (part (b))?
Hint. If, as you surmise, on any given day, the concierge replaces the bulb at rate $0.01$, and the maintenance employee replaces the bulb at rate $0.02$, how many times more likely is the maintenance employee to replace the bulb than the concierge?
And since one of them must eventually replace any given bulb, what is the probability that it is the concierge?