  Let $\mathfrak{S}_n$ and $\mathfrak{B}_n$ denote the respective sets of
ordinary and bigrassmannian (BG) permutations of order $n$, and let
$(\mathfrak{S}_n,\leq)$ denote the Bruhat ordering permutation poset. We study
the restricted poset $(\mathfrak{B}_n,\leq)$, first providing a simple
criterion for comparability. This criterion is used to show that that the poset
is connected, to enumerate the saturated chains between elements, and to
enumerate the number of maximal elements below $r$ fixed elements. It also
quickly produces formulas for $\beta(\omega)$ ($\alpha(\omega)$ respectively),
the number of BG permutations weakly below (weakly above respectively) a fixed
$\omega\in\mathfrak{B}_n$, and is used to compute the M\"obius function on any
interval in $\mathfrak{B}_n$.
  We then turn to a probabilistic study of $\beta=\beta(\omega)$
($\alpha=\alpha(\omega)$ respectively) for the uniformly random
$\omega\in\mathfrak{B}_n$. We show that $\alpha$ and $\beta$ are
equidistributed, and that $\beta$ is of the same order as its expectation with
high probability, but fails to concentrate about its mean. This latter fact
derives from the limiting distribution of $\beta/n^3$.
  We also compute the probability that randomly chosen BG permutations form a
2- or 3-element multichain.
