The problem:
There is a system with a clock ticking at a constant rate measured every second.
Within this system there are n loops that are ongoing. The period of the loop is between 80 and 120 seconds (inclusive both), every time a single loop is completed the period is again set to a random value between 80 and 120 seconds.
If the loop is completed during any one second that second is considered a "success".
The question: What is the probability that given n loops, there will be m = 20 seconds in a row that do not "succeed" at least once during any one random 20 second period picked at random.
My attempt / thinking:
Since the period is random, even if regular, once the system comes to steady state it will result in an evenly distributed random system for any time period below 80 seconds.
The average period is 100 seconds, so the average probability of any one second having a success from n = 1 will be: $$\frac{1}{100}$$
The probability of failure per second given n loops is: $$\left(1-\frac{1}{100}\right)^n$$
The probability of success per second given n loops is: $$1-\left(1-\frac{1}{100}\right)^n$$
Would the probability for at at least 1 success for m = 20 seconds be: $$1-\left(\left((1-\frac{1}{100}\right)^{n}\right)^{m} = 1-\left(\left((1-\frac{1}{100}\right)^{n}\right)^{20}$$
Or is this oversimplifying the problem? Is my assumption about the period distributing into a totally random distribution valid or am I missing something?
Thank you for your help! I have been banging my head against this one for a little while. It does not come from a textbook but a real life situation so you if need clarification please let me know.