# An Elementary Analysis of the Probability That a Binomial Random Variable Exceeds Its Expectation

arXiv: https://arxiv.org/abs/1712.00519 · HTML: https://mlsanity.com/paper/1712.00519/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Dec 2017.

## Built on by (most cited first, 2 of 2)

- [Stochastic Shortest Path with Sparse Adversarial Costs](https://mlsanity.com/paper/2511.00637/) (2025)
- [Improving Generalization Bounds for VC Classes Using the Hypergeometric Tail Inversion](https://mlsanity.com/paper/2111.00062/) (2021)

## Abstract (arXiv)

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.