Problem 151


Problem 151


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.


标准大小的纸张:有关期望的问题

一家打印店每周有16份打印工作,每份打印工作需要一张A5大小的特殊彩色打样纸。

每周一早晨,领班会打开一个新信封,其中有一张A1大小的特殊彩色打样纸。

他将这张纸一裁为二,得到两张A2大小的纸。再将其中一张一裁为二,得到A3大小的纸,如此直到他裁出一张用于第一份打印工作的A5大小的彩色打样纸。

所有剩下的纸张将会重新放回信封里。

接下来的每次打印工作前,领班会从信封里拿出随机拿出一张纸,如果恰好是A5大小的,他会直接拿去使用,否则他会重复一裁为二的过程直到他得到一张A5大小的纸,并将剩下的纸重新放回信封里。

除了每周的第一次和最后一次打印工作外,求在这周当中领班在拿纸时发现信封里只有一张纸的次数的期望值。

你的答案应当保留六位小数,即以如下的形式x.xxxxxx。