I have a machine part that have lifetime uniformly distributed between 0 years and 1 year. Whenever a part fails, it is immediately replaced with a new identical part. I know that lifetimes of replacement parts are independent.
My question is: what is the probability that 5 years from now, I replace the part exactly 8 times.
I know that the expected lifetime of a bulb is 1/2 years, but I am not sure how to incorporate that into calculating the probability.
One of the approaches I am thinking of is to model the question into a Markov Chain with two states, Good and Bad. I know that P(Bad, Good) = 1 but I am not sure what the transition probabilities starting from state Good would look like?
Maybe I am complicating the problem too much, any help would be appreciated.