  For a given hypergraph $H$ and a vertex $v\in V(H)$, consider a random
matching $M$ chosen uniformly from the set of all matchings in $H.$ In $1995,$
Kahn conjectured that if $H$ is a $d$-regular linear $k$-uniform hypergraph,
the probability that $M$ does not cover $v$ is $(1 + o_d(1))d^{-1/k}$ for all
vertices $v\in V(H).$ This conjecture was proved for $k = 2$ by Kahn and Kim in
$1998.$
  In this paper, we disprove this conjecture for all $k \geq 3.$ For infinitely
many values of $d,$ we construct $d$-regular linear $k$-uniform hypergraph $H$
containing two vertices $v_1$ and $v_2$ such that $\mathcal{P}(v_1 \notin M) =
1 - \frac{(1 + o_d(1))}{d^{k-2}}$ and $\mathcal{P}(v_2 \notin M) = \frac{(1 +
o_d(1))}{d+1}.$ The gap between $\mathcal{P}(v_1 \notin M)$ and
$\mathcal{P}(v_2 \notin M)$ in this $H$ is best possible. In the course of
proving this, we also prove a hypergraph analog of Godsil's result on matching
polynomials and paths in graphs, which is of independent interest.
