Assume $X$ is discrete random variable. It has some distribution on integers $0,1,...,m$
Then is $P(X>k | X=m) \ge P(X>k)$ true? If it is, how to prove it in a rigorous way?
For me, it is tempting to think that it is true because given $X$ taking the maximum value, the probability of $X>k$ should be higher. But I can't figure out how to prove my thought.