  The classical Erd\H os-Ko-Rado (EKR) Theorem states that if we choose a
family of subsets, each of size (k), from a fixed set of size (n (n > 2k)),
then the largest possible pairwise intersecting family has size (t ={n-1\choose
k-1}). We consider the probability that a randomly selected family of size
(t=t_n) has the EKR property (pairwise nonempty intersection) as $n$ and
$k=k_n$ tend to infinity, the latter at a specific rate. As $t$ gets large, the
EKR property is less likely to occur, while as $t$ gets smaller, the EKR
property is satisfied with high probability. We derive the threshold value for
$t$ using Janson's inequality. Using the Stein-Chen method we show that the
distribution of $X_0$, defined as the number of disjoint pairs of subsets in
our family, can be approximated by a Poisson distribution. We extend our
results to yield similar conclusions for $X_i$, the number of pairs of subsets
that overlap in exactly $i$ elements. Finally, we show that the joint
distribution $(X_0, X_1, ..., X_b)$ can be approximated by a multidimensional
Poisson vector with independent components.
