Erdős Problem #525 — Minimum modulus of almost every sign polynomial

Among the $2^{n+1}$ polynomials $P(z)=∑_{k=0}^n ε_kz^k$ with $ε_k∈{-1,1}$, do all but $o(2^n)$ satisfy $\min_{|z|=1}|P(z)|<1$? More generally, determine the typical size of this minimum.