I don't get it. The probability must depend on how likely I am to select any specific integer, i.e. probability mass function of N. The probability cannot depend on the value of the random variable itself but could involve any parameters that define its distribution.
I would consider something like the following a valid question:
"Let N be a random integer between 0 and M-1 with uniform distribution. What is the probability that N is even?"
Then an answer could be "the probability that N is even is 1/2 if M is even and (M+1)/(2M) if M is odd". See this does not involve N but does involve M which is a parameter for the distribution of N.
Your explanation seems to invoke some "common sense" which I am not able to unify with my understanding of probability theory.
One meaning is that summing the probabilities for N=1,2,3, ... and checking how many numbers actually meet the criteria yield the same result in the limit. More precisely, their ratio goes to 1 as N goes to infinity.
Edit: define f(N) as the number of numbers below N that match. Consider the function f(N)/N. We call a function of N the probability of N matching if it asymptomatically approaches f(N)/N.
You are thinking "given N, what is the probability that N is a perfect nth power?". Think about it in the reverse. "given N, what is the probability that the nth root of N is an integer". For a whole bunch of different Ns, we will get some real number between 0 and 1.
To match up to your example. "For a number M, m is the remainder of M: m = M - floor(M), what is the probability that m is zero given M"
Intuitively, you can think of N as a rough measure of the "ballpark" around where you are in the number line. As you move up to higher numbers, perfect powers get more and more scarce, so the probability of finding one by picking a number at random becomes lower.
Imagine you don't have an easy way to solve the equation x^n = N where x is an integer. Yet, for a given N you want to find a probability that such x (solution) exists. There is nothing here about how you select N.
I would consider something like the following a valid question: "Let N be a random integer between 0 and M-1 with uniform distribution. What is the probability that N is even?"
Then an answer could be "the probability that N is even is 1/2 if M is even and (M+1)/(2M) if M is odd". See this does not involve N but does involve M which is a parameter for the distribution of N.
Your explanation seems to invoke some "common sense" which I am not able to unify with my understanding of probability theory.