Sam Gutmann - Department of Mathematics, Northeastern University, ۳۶۰ Huntington Ave, Boston, MA, USA.
Mark Mixer - School of Computing and Data Science, Wentworth Institute of Technology, ۵۵۰ Huntington Ave, Boston, MA, USA.
Steven Morrow - School of Computing and Data Science, Wentworth Institute of Technology, ۵۵۰ Huntington Ave, Boston, MA, USA.

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.

derangement, Fixed Point, probability