Hoeffding lower-tail bound for Hamming balls ($2^{-n}|B_{an}|\le e^{-2n(1/2-a{)}^2}$)

For a uniform vertex of the Hamming cube, distance from a fixed vertex is the sum of $n$ independent Bernoulli random variables with parameter $1/2$. The lower-tail Hoeffding inequality implies that the fraction of vertices at distance at most $an$, $0<a<1/2$, is at most $e^{-2n(1/2-a{)}^2}$. Strict-radius balls satisfy the same upper bound.