  Let $f_\omega(z)=\sum\limits_{j=0}^{\infty}\chi_j(\omega) a_j z^j$ be a
random entire function, where $\chi_j(\omega)$ are independent and identically
distributed random variables defined on a probability space $(\Omega,
\mathcal{F}, \mu)$. In this paper, we first define a family of random entire
functions, which includes Gaussian, Rademacher, Steinhaus entire functions.
Then, we prove that, for almost all functions in the family and for any
constant $C>1$, there exist a constant $r_0=r_0(\omega)$ and a set $E\subset
[e, \infty)$ of finite logarithmic measure such that, for $r>r_0$ and $r\notin
E$, $$ |\log M(r, f)- N(r,0, f_\omega)|\le
(C/A)^{\frac1{B}}\log^{\frac1{B}}\log M(r,f) +\log\log M(r, f), \qquad a.s. $$
where $A, B$ are constants, $M(r, f)$ is the maximum modulus, and $N(r, 0, f)$
is the weighted counting-zero function of $f$. As a by-product of our main
results, we prove Nevanlinna's second main theorem for random entire functions.
Thus, the characteristic function of almost all functions in the family is
bounded above by a weighed counting function, rather than by two weighted
counting functions in the classical Nevanlinna theory. For instance, we show
that, for almost all Gaussian entire functions $f_\omega$ and for any
$\epsilon>0$, there is $r_0$ such that, for $r>r_0$, $$ T(r, f) \le N(r,0,
f_\omega)+(\frac12+\epsilon) \log T(r, f). $$
