An elementary analysis of the probability that a binomial random variable exceeds its expectation

We give an elementary proof of the fact that a binomial random variable X with parameters n and 0.29∕n≤p<1 with probability at least 1∕4 strictly exceeds its expectation. We also show that for 1∕n≤p<1−1∕n, X exceeds its expectation by more than one with probability at least 0.0370. Both probabilities approach 1∕2 when np and n(1−p) tend to infinity.