  We study the probability $p \equiv p_\eta(t)$ that two directed polymers in a
given random potential $\eta$ and with fixed and nearby endpoints, do not cross
until time $t$. This probability is itself a random variable (over samples
$\eta$) which, as we show, acquires a very broad probability distribution at
large time. In particular the moments of $p$ are found to be dominated by
atypical samples where $p$ is of order unity. Building on a formula established
by us in a previous work using nested Bethe Ansatz and Macdonald process
methods, we obtain analytically the leading large time behavior of {\it all
moments} $\overline{p^m}\simeq \gamma_m/t$. From this, we extract the exact
tail $\sim \rho(p)/t$ of the probability distribution of the non-crossing
probability at large time. The exact formula is compared to numerical
simulations, with excellent agreement.
