conditional probability of derangements and fixed points

The probability that a random permutation in s_n is a derangement is well known to be displaystylesumlimits_{j=0}^n (-1)^j frac{1}{j!}. in this paper, we consider the conditional probability that the (k+1)^{st} point is fixed, given there are no fixed points in the first k points. we prove that when n neq 3 and k neq 1, this probability is a decreasing function of both k and n. furthermore, it is proved that this conditional probability is well approximated by $frac{1}{n} - frac{k}{n^2(n-1)}. similar results are also obtained about the more general conditional probability that the (k+1)^{st} point is fixed, given that there are exactly d fixed points in the first k points.