We study a family of random Taylor series $$F(z) = \sum_{n\ge 0} ζ_n a_n z^n$$ with radius of convergence almost surely $1$ and independent identically distributed complex Gaussian coefficients $(ζ_n)$; these Taylor series are distinguished by the invariance of their zero sets with respect to isometries of the unit disk. We find reasonably tight upper and lower bounds on the probability that $F$ does not vanish in the disk $\{|z|\le r\}$ as $r\uparrow 1$. Our bounds take different forms according to whether the non-random coefficients $(a_n)$ grow, decay or remain of the same order. The results apply more generally to a class of Gaussian Taylor series whose coefficients $(a_n)$ display power-law behavior.