The probability \(\operatorname{P}_H(r)\) of the event that a function \(f\) has no zeros in the disc \(\{|z| < r\}\) is called the hole probability. This paper is concerned with a random entire function \[ f(z) = \sum\limits_{n = 0}^\infty {{\phi _n}{a_n}{z^n}}, \] where the \(\phi_n\) are independent standard complex Gaussian coefficients, and the \(a_n\) are positive constants, which satisfy \[ \mathop {\lim }\limits_{x \to \infty }\frac{\log {a_n}}{n} = - \infty. \] When \(r\) grows to infinity, the decay rate of the hole probability is studied. The author continues the work started in [Int. Math. Res. Not. 2010, No. 15, 2925–2946 (2010; Zbl 1204.60043)], using the classical Wiman-Valirom theory of the growth of power series. Assuming that the sequence \(a_n\) is logarithmically concave, upper and lower bounds for \(\operatorname{P}_H(r)\) are found which help in proving the main result about hole probability (Theorems 1 and 2).