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.