Probability of a Gaussian process indexed by a random variable with a known distribution.

Question

A Gaussian process $y(x)$ is indexed by a random variable $X$. If we know the distribution of $X$, the mean function and covariance functions of the Gaussian process (which depend on $X$), what is the probability $\Pr\{y(x)<c, \exists x \in R_X\}$ where $c$ is a constant and $R_x$ is the support of $X$?