An Easy Proof of a Weak Version of Chernoff inequality

TL;DR

This paper presents a simple proof of a weak form of Chernoff inequality, showing that the probability of getting at most half the heads in 6M coin flips is at most 1/2^M.

Contribution

It provides an easy and straightforward proof of a weaker version of Chernoff inequality, simplifying understanding of tail bounds for coin flips.

Findings

Probability of at most M heads in 6M flips ≤ 1/2^M

Simplified proof technique for a weak Chernoff bound

Accessible approach to tail probability estimation

Abstract

We prove an easy but very weak version of Chernoff inequality. Namely, that the probability that in $6M$ throws of a fair coin, one gets at most $M$ heads is $\leq 1/2^M$.