  We show that for Gaussian random SU(2)polynomials of a large degree $N$ the
probability that there are no zeros in the disk of radius $r$ is less than
$e^{-c_{1,r} N^2}$, and is also greater than $e^{-c_{2,r} N^2}$. Enroute to
this result, we also derive a more general result: probability estimates for
the event that the number of complex zeros of a random polynomial of high
degree deviates significantly from its mean.
