  We answer a question of Ahlberg and Steif (2014) by finding the tail
behaviour of the crossing probability in near-critical planar percolation.
Interestingly, this superexponentially small behaviour is different from the
case of dynamical percolation, where the analogous tail probability was proved
to be at least exponential and at most superpolynomial by Hammond, Mossel and
Pete (2012). The proof is simple, given the scale covariance established by
Garban, Pete and Schramm (2013).
