The Fluid Dynamics of Pill Popping

This is a very silly question that I've been obsessing about for weeks. It's possible I've even written about it before, though I can't find any trace of that. It's a question that simultaneously feels so basic that science has to have an idea about it, but also I can't think of a single practical application that would prompt anyone to study it.

Let me lay out the scene.

I have a bottle of pills, from which I take one pill each day. To do this, I tilt the pill bottle and "pour" one pill out onto my hand, then return the bottle upright and place it back on the shelf. Other than this, I don't shake or mix the pill bottle in any way.

When the pill bottle is close to -- but not quite -- empty, I get a new bottle. At this point, I pour the remaining pills from the old bottle into the new bottle (again, without otherwise mixing or shaking). So if I had 10 pills left from the old bottle, and 90 pills in the new bottle, the new bottle now has 100 pills (including the 10 "old" pills poured over the top). Then I start the cycle again of taking one pill a day.

Here's my question: on average, how long do we predict it would take me to consume all of the "old" pills (assuming I don't vary my routine)?

One answer is that the order of the pills being poured out is essentially random (I have an equal chance of "selecting" any given pill), and so the answer of how long it will take me to pour out the ten old pills is the same as the answer for any randomly selected ten pills. But it seems wrong to suggest that the order is in fact random -- the fact that these pills were specifically placed on the top of the pile of pills should mean that they have a higher likelihood of being poured out first (right?).

So another answer at the opposite end of the spectrum would be that since the old pills are at the top of the pill bottle, they should be the first ten pills that I consume (or close to it). Something like that is the intention of pouring them onto the top of the pile. But this also strikes me as unlikely -- intuitively, I feel like the act of pouring does not necessarily result in the "top" pills necessarily being poured out first. It does some mixing on its own. More broadly, when I imagine the physical act of the pile of pills cascading down the side of the pill bottle into my hand, it's very easy for me to imagine pills that were not on top skipping ahead and getting into my hand first.

In short, I suspect that I should pour the "old" pills more quickly than the new ones by some indeterminate factor -- more quickly than random selection, less quickly than "they'll be the first ten". It's a question, in essence, about the "fluid dynamics" of pills, which is a concept that tickles me for some reason.

This actually would be pretty easy to measure in concept: give each pill a number, instruct research subjects on my routine, and then have them mark down the number of the pill they pour out each day. But has anyone actually investigated this? On the one hand, it feels like utterly pointless knowledge. On the other hand, scientists love finding out about the properties of random subjects!

Anyway, for anyone working in a germane field, this is a free research proposal for you. Have at it.