I wanted to get quarters out of a change jar and remembered exactly this. Just shook it around (lightly) until there were quarters at the top.

Bingo. I read the code to be sure there wasn't a factor to model this. The answer should depend on the relative probability of choosing a whole pill or a half pill. The code assumes equal odds, with no state dependence on the remaining ratio.

I was startled by the formula, though. It's the same as the Gilbert-Shannon-Reeds model for card shuffling, where the probability of a card falling from each packet scales to the number of cards left in each packet. (I wrote the "seven shuffles" paper with Persi Diaconis.)

> The code assumes equal odds

This is stated explicitly in the problem: Each day you select a pill at random

Clearly stating that the model is unrealistic doesn't make the model realistic.

On the other hand, you always put the half pills you create on top of the pills already in there, not in a random spot in the bottle.