University of IsfahanTransactions on Combinatorics2251-865712120230301Conditional probability of derangements and fixed points11262629110.22108/toc.2022.131705.1941ENSamGutmannDepartment of Mathematics, Northeastern University, 360 Huntington Ave, Boston, MA, USA.Mark D.MixerSchool of Computing and Data Science, Wentworth Institute of Technology, 550 Huntington Ave, Boston, MA, USA.StevenMorrowSchool of Computing and Data Science, Wentworth Institute of Technology, 550 Huntington Ave, Boston, MA, USA.Journal Article20211201The 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.https://toc.ui.ac.ir/article_26291_f825454f21ba5f39cbf58e4058b7d906.pdf