How often can a linear speed sort succeed?
David Eppstein gave away the answer: an $O(n)$ comparison-based algorithm will succeed on a negligible fraction of permutations. One way to see that is that an $O(n)$ algorithm can only take care of $2^{O(n)}$ different input permutations, compared to the $n! = 2^{\Omega(n\log n)}$ possible ones.