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=۰}^n (-۱)^j \frac{۱}{j!}. In this paper, we consider the conditional probability that the (k+۱)^{st} point is fixed, given there are no fixed points in the first k points. We prove that when n \neq ۳ and k \neq ۱, this probability is a decreasing function of both k and n. Furthermore, it is proved that this conditional probability is well approximated by \frac{۱}{n} - \frac{k}{n^۲(n-۱)}. Similar results are also obtained about the more general conditional probability that the (k+۱)^{st} point is fixed, given that there are exactly d fixed points in the first k points.

Keywords:

derangement , Fixed Point , probability