  Based on the properties of probability distributions of functions of random
variables, we proposed earlier a simple stringy mechanism that prefers the
meta-stable vacua with a small cosmological constant \Lambda. As an
illustration of this approach, we study in this paper particularly simple but
non-trivial models of the K\"ahler uplift in the large volume flux
compactification scenario in Type IIB string theory, where all parameters
introduced in the model are treated either as fixed constants motivated by
physics, or as random variables with some given uniform probability
distributions. We determine the value w_0 of the superpotential W_0 at the
supersymmetric minima, and find that the resulting probability distribution
P(w_0) peaks at w_0=0; furthermore, this peaking behavior strengthens as the
number of complex structure moduli increases. The resulting probability
distribution P(\Lambda) for meta-stable vacua also peaks as \Lambda -> 0, for
both positive and negative \Lambda. This peaking/divergent behavior of
P(\Lambda) strengthens as the number of moduli increases. In some scenarios for
\Lambda > 0, the likely value of \Lambda decreases exponentially as the number
of moduli increases. The light cosmological moduli issue accompanying a very
small \Lambda is also mentioned.
