Real-world problem: Behavior of a conditional probability

Question

Apologies for the rather vague phrasing of the title; I'm trying to formulate this problem as exactly as I can and perhaps not succeeding very well.

The problem is inspired by a real-world situation: For a medical condition, my doctor has prescribed me $75$ pills, to be taken over $30$ days, $2.5$ pills once per day. The pharmacy has dispensed me $75$ whole pills. My doctor has assured me that, rather than asking the pharmacy to dispense smaller pills, I can simply split a whole pill (and return the unused half pill to the bottle).

I've developed the following method of getting the pills from the pill bottle:

1. Take one item, that is, one pill or half pill, from the bottle at random.
2. Repeat just until I have enough pills (or perhaps just more than enough) to take my $2.5$ pills worth of medication. In particular, repeat until I have one of the following:
   
   • $5$ half pills
   • $4$ half pills and $1$ whole pill
   • $3$ half pills and $1$ whole pill
   • $2$ half pills and $2$ whole pills
   • $1$ half pill and $2$ whole pills
   • $0$ half pills and $3$ whole pills
3. In any of the first five cases above, take exactly $2.5$ pills worth of medication, replacing any extra half pill in the bottle. In the last case ($3$ whole pills) split one of the pills, take the $2.5$ pills, and replace the extra half pill.

The question I have, based on my real-world observation of the pill bottle is:

How is the proportion of half pills in the bottle expected to behave over time? Does this change in general if I have $5n$ pills to consume over $2n$ days, as $n$ increases without bound?

Obviously near the beginning, there will be very few half pills. But as I tried to evaluate (for example) the probability that there would be three half pills after $6$ days, I just got very confused. And as the number of days approached $30$ the number of different possibilities grew.

Answer 1

A very similar problem ($1/2$ pill instead of $5/2$ pills) is discussed in A Drug-Induced Random Walk by Daniel J. Velleman.

I have the PDF but I am not sure how to insert it here (it is not a picture).