Problem 740
Secret Santa
Secret Santa is a process that allows $n$ people to give each other presents, so that each person gives a single present and receives a single present. At the beginning each of the $n$ people write their name on a slip of paper and put the slip into a hat. Each person takes a random slip from the hat. If the slip has their name they draw another random slip from the hat and then put the slip with their name back into the hat. At the end everyone buys a Christmas present for the person whose name is on the slip they are holding. This process will fail if the last person draws their own name.
In this variation each of the $n$ people gives and receives two presents. At the beginning each of the $n$ people writes their name on two slips of paper and puts the slips into a hat (there will be $2n$ slips of paper in the hat). As before each person takes from the hat a random slip that does not contain their own name. Then the same person repeats this process thus ending up with two slips, neither of which contains that person’s own name. Then the next person draws two slips in the same way, and so on. The process will fail if the last person gets at least one slip with their own name.
Define $q(n)$ to be the probability of this happening. You are given $q(3) = 0.3611111111$ and $q(5) = 0.2476095994$ both rounded to $10$ decimal places.
Find $q(100)$ rounded to $10$ decimal places.
神秘圣诞老人
神秘圣诞老人是一种$n$个人互送圣诞礼物的节日传统活动，其中每个人送出一份礼物并收到一份礼物。这个活动的流程如下：首先，$n$个参与活动的人各自在一张纸条上写上自己的名字，然后将所有这些纸条放进一个帽子里；然后，每个人依次从帽子中随机抽出一张纸条，若抽到自己的名字，则把纸条放回去并重新再抽；最后，每个人买一份圣诞礼物，送给自己抽到的纸条上写名字的那个人。可以看出，如果最后一个抽纸条的人不幸抽到了自己的名字，这个活动就失败了。
考虑上述活动的一个变种，其中每个人送出并收到两份礼物。首先，$n$个参与活动的人各自在两张纸条上写上自己的名字，然后将所有这些纸条放进一个帽子里（这样一来帽子中总共有$2n$张纸条）。然后，每个人依次从帽子中随机抽出两张纸条，若抽到自己的名字，则把纸条放回去重新再抽。在该变种中，如果最后一个抽纸条的人抽到至少一张有自己名字的纸条，就认为这个活动失败了。
记$q(n)$为活动失败的概率。已知$q(3) = 0.3611111111$，$q(5) = 0.2476095994$，均保留小数点后$10$位小数。
求$q(100)$，并保留小数点后$10$位小数。