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. (C) 2018 Elsevier B.V. All rights reserved.