雕像造型
Figurine Figuring
题目详情
Jane 在这个假期收到了 78 个雕像作为礼物：12 个鼓手 击鼓、11 个风笛手、10 个跳跃领主等，减少到 1 个 梨树上的鹧鸪。它们都混合在一起装在一个大袋子里。 她同意她的朋友亚历克斯的观点，这似乎太多了 雕像是一个人才能拥有的，所以她决定把自己的一些送给别人 亚历克斯的雕像。简会统一随机拉出小雕像 一次一个袋子，直到她从梨中取出鹧鸪 树，并将把她从树上拉出来的所有雕像都给亚历克斯 袋子（除了鹧鸪，那是简的最爱）。
如果n是Alex获得的任何一种装饰品的最大数量， n 的期望值是多少（精确到七位有效数字）？
Jane received 78 figurines as gifts this holiday season: 12 drummers drumming, 11 pipers piping, 10 lords a-leaping, etc., down to 1 partridge in a pear tree. They are all mixed together in a big bag. She agrees with her friend Alex that this seems like too many figurines for one person to have, so she decides to give some of her figurines to Alex. Jane will uniformly randomly pull figurines out of the bag one at a time until she pulls out the partridge in a pear tree, and will give Alex all of the figurines she pulled out of the bag (except the partridge, that’s Jane’s favorite).
If n is the maximum number of any one type of ornament that Alex gets, what is the expected value of n, to seven significant figures?
解析
本月的难题需要仔细计算才能解决 易于处理。一种可行的方法：通过跟踪 鹧鸪雕像的位置和相同数量的最大数量 在它面前绘制的雕像为十二套不同的雕像 被插入到排列中，将空间减少到矩阵中 尺寸 78 x 12。最终预期值约为 6.859787。
恭喜所有找到预期值在范围内的人 要求的准确性。
Original Explanation
This month’s puzzle required a careful computation to remain tractable. One approach that would work: by keeping track of the position of the partridge figurine and the maximum count of identical figurines drawn before it as the twelve different sets of figurines were inserted into the permutation reduced the space into a matrix of size 78 x 12. The final expected value was approximately 6.859787.
Congratulations to everyone who found the expected value to within the requested accuracy.