The probabilistic interpretation of ramanujan's constant $ e^{\pi\sqrt{163}}$

Question

Some of the mathematical constants have interesting probabilistic interpretations. For example,

1. "$\pi$". Suppose two integers are chosen at random. What is the probability that they are comprime, that is, have no common factor exceeding 1? The answer is $\large\frac{6}{\large\pi^2}$

2. "Apery's constant". Given three random integers, the probability that no factor exceeding 1 divides them all is $\large\frac{1}{\zeta(3)}$

Does there exist probabilistic interpretation of Ramanujan's constant $\large e^{\large\pi\large\sqrt{163}}$ ?