题目151：标准大小的纸张：一个期望问题

欧拉计划

Paper sheets of standard sizes: an expected-value problem

A printing shop runs 16 batches (jobs) every week and each batch requires a sheet of special colour-proofing paper of size A5.

Every Monday morning, the foreman opens a new envelope, containing a large sheet of the special paper with size A1.

He proceeds to cut it in half, thus getting two sheets of size A2. Then he cuts one of them in half to get two sheets of size A3 and so on until he obtains the A5-size sheet needed for the first batch of the week.

All the unused sheets are placed back in the envelope.

At the beginning of each subsequent batch, he takes from the envelope one sheet of paper at random. If it is of size A5, he uses it. If it is larger, he repeats the 'cut-in-half' procedure until he has what he needs and any remaining sheets are always placed back in the envelope.

Excluding the first and last batch of the week, find the expected number of times (during each week) that the foreman finds a single sheet of paper in the envelope.

Give your answer rounded to six decimal places using the format x.xxxxxx .

题目：

一个印刷车间每周都要完成 16 批工作，每批工作都需要一些 A5 大小、经过特殊着色处理的纸张。
每周一的早上，工头打开一个新的包装袋，里面有着一大张，A1 大小的原料纸张。

他开始把它切半，得到两张 A2 大小的纸，接着，将其中之一再切半成 A3 大小，如此继续进行，直到他得到本周第一批工作需要的 A5 大小的原料纸。

所有没用到的又放回包装袋中。

在每个子作业的开始前，他会随机从包装袋中抽一张，如果正好 A5 大小，就拿来用。如果大了，他会继续进行对半切，直到他得到他想要的，再把其它的放回去。

除了每周第一批和最后一批的任务，求每周工头从包装袋中发现只有一张纸（任意大小）的次数的期望值，

将答案取六位小数，格式为X.XXXXXX

加餐

直接dp计算，或者模拟随机过程暴力拟合概率都可