In a city there's $a$ road lamps. the life time of each lamp is exponential with expected value of $b$ months and the life times of different lamps are independent.
In the start of the year, all the lamps are working, and whenever a lamp breaks, we change it immediately with a new lamp.
What is the probability that no lamp breaks between the hours $2:00$ and $3:00$, in the morning of the first day of february? (you can assume that in each month there's $30$ days).
What I immediately got after reading the question, that the time between each lamp break, is just a random variable which is the minimum of all the exponentials of every lamp, and that is known to have distribution of $exp(a\frac{1}{b})$. (sum of all the expected values of all the lamps), and let's call it $X$.
Now what I want to calculate is the probability of no lamp breaking inbetween $2-3$, but first I converted them to months, which gives us $\frac{2}{720},\frac{3}{720}$.
Now I really want to say that from memorylessness property, we can assume that $\frac{2}{720}$ is our start, and we need $X>\frac{1}{720}$ in order to have no lamps burnt in between these two hours.
But for some reason it doesn't feel alright with me, I know that for memorylessness: $P(X>s+t|X>t)=P(X>s)$.
And for example $E(X+Y|X>Y)=E(X)$, or in simple words, the probability of my random variable $X$ being greater than something knowing that my random variable is greater than $a$, is the same as the probability of my random variable being greater than something$-a$.
I'm not sure how to fit my understanding of memorylessness property (if I really understand it..) into this question, what extra information I know about my random variable $X$ that I could use this property here?
I would really appreciate any help in how to connect my understanding of memorylessness property to this question, or pointing out what's wrong with my understanding of memorylessness property if I understood it wrong.