Conditional probability of derangements and fixed points

The probability that a random permutation in S_n is a derangement is well known to be \displaystyle\sum\limits_{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.