I have a bottle of 100 pills. The daily dose is 1/2 pill, so if the first pill I extract is a whole pill, I split it and put 1/2 back.
Just out of my own general curiosity, I'd like to model the probability of extracting a whole pill vs. a half pill over time, but I'm not sure how to start.
I have written a paper on this which will be published in the American Mathematical Monthly sometime in the next year or so. The title is "A drug-induced random walk." The main theorem is this: Consider a bottle of $n$ pills. Every day, you remove a pill from the bottle at random (with each pill equally likely to be chosen). If it is a whole pill, you cut it in half, take half of the pill, and return the other half to the bottle. If it is a half pill, then you take it and nothing is returned to the bottle. At any time, let $x$ be the fraction of the original pills in the bottle that are still whole, and let $y$ be the fraction that are now half pills. ($x+y$ may be less than $1$, since some pills may have been used up completely.) Then the point $(x,y)$ executes a random walk in the plane, starting at the point $(1,0)$ (all pills whole) and ending at $(0,0)$ (no pills left). The theorem says that for large $n$, the random walk will approximately follow the curve $y = -x \ln x$. More precisely, the theorem says that for every $\epsilon > 0$, the probability that the walk stays within $\epsilon$ of the curve $y = -x \ln x$ approaches $1$ as $n$ approaches infinity. The paper also answers the questions "What is the expected number of whole pills removed before the first half pill is removed?" and "What is the expected number of half pills removed after the last whole pill is removed?"
