A Preference for A5

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 supervisor opens a new envelope, containing a large sheet of the special paper with size A1.
The supervisor proceeds to cut it in half, thus getting two sheets of size A2. Then one of the sheets is cut in half to get two sheets of size A3 and so on until an A5-size sheet is obtained, which is 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, the supervisor takes from the envelope one sheet of paper at random. If it is of size A5, then it is used. If it is larger, then the ‘cut-in-half’ procedure is repeated until an A5-size sheet is obtained, 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 supervisor finds a single sheet of paper in the envelope.
Give your answer rounded to six decimal places using the format x.xxxxxx .

The problem is the find the expected number of times during each week that the supervisor finds a single sheet of paper in the envelope.

First, let me explain paper sizes. For example, an A4 size paper is half the size of A3, an A5 is half the size of A4, and an A2 is half the size of A1. So in the beginning, the A1 paper is cut into halves to get A2 and so forth as stated in the problem.

The total number of jobs/batches during the week are 16. However, we are being asked to exclude the first and the last one. This means we will only consider 16 – 1 – 1 = 14 batches.

From starting from an A1 paper, when it is first cut in half, an A2 and two A3s are generated, but one of the A3s is used immediately (for the first batch/job). Next, what’s left of the A2 is cut in half to generate an A3. Now, the untouched A2 is cut in half to generate two A3s. Then one of these A3s is cut in half to produce two A4s. From here, one A4 is used and the other is cut in half to generate two A5s.

Understand that at each stage, there are two possibilities. Either there are two sheets of the next smaller paper or only one. If there’s only one, the other one has been used and a sheet from the pile of the next larger paper in size is cut in this stage. Otherwise, the leftover sheets from the previous stage are used until exhausted before the cut-in-half procedure is repeated.

Now the situation described in the problem happens when only one sheet of paper is left in the envelope. So we need to look for the times a cut-in-half procedure is repeated, indicating that all the previous sheets of smaller sizes had been exhausted (and thus there was only one sheet remaining). This happens in the following cases, starting from the second job:

2nd jobs: A3 -> A4 -> A5 (A2s have been exhausted)

5th jobs: A2 -> A3 -> A4 -> A5 (A1s have been exhausted)

10th jobs: A1 -> A2 -> A3 -> A4 -> A5 (Restart, all papers are exhausted from previous jobs)

Each time the cycle restarts, it’s because all the paper has been used up and we have to begin cutting from the beginning, starting from A1.

On Job 2, the supervisor finds a single sheet (leading to a cut) 0 times.
On Job 5, the supervisor finds a single sheet (leading to a cut) 1 time.
On Job 10, the supervisor finds a single sheet (leading to a cut) 2 times.

This forms a sequence of jobs where the supervisor finds a single sheet and needs to cut them in halves. To find the expected value, we multiply the number of times the supervisor finds a single sheet (0,1,2) with the number of jobs remaining, then sum these values, and divide this by the total number of jobs (excluding the first and last one which is 14).

E[k] = (12/14)*0 + (7/14)*1 + (4/14)*2 = 0/14 + 7/14 + 8/14 = 15/14

Thus the expected number of times the supervisor finds a single sheet of paper in the envelope per week (excluding the first and last batch of the week) is approximately 1.071429. Rounded to six decimal places, the answer is 1.071429.

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