Suppose we have a random variable X with binomial distribution B(n, p). I'm interested in the probability P(X ≥ E[X]). In particular, is there some inequality for how this probability changes as n and p grow? Is there some lower and upper bound for this probability? I was not able to find any resource on this topic.
For context, I am using some probabilistic computational model and I'm trying to see how it behaves as n (the size of the input) grows.
