# The End of the Bottle

At my doctor’s direction, I take a vitamin D supplement everyday. Specifically, I take two pills everyday. Each bottle is supposed to have 360 pills. So on each day there should be an even number of pills in the bottle and on the last day, there should be exactly two for me to take. This time, when I got to the end of the bottle, there was only one. Did I make a mistake and not take two each day, or did the bottle come with an incorrect number of pills?

There were two assumptions that led to my expectation that there would be two pills instead of one left on the last day. One assumption was that I took exactly two pills each day with no loss. The other assumption was that the bottle did indeed have exactly 360 pills to start. At least one of these assumptions must have been false to have led to the outcome of one pill on the last day. It is regrettably impossible to know which one. That means my question is impossible to answer, at least with total certainly. I could try to predict the likelihood of each scenario.

There are ways that I could estimate the probability that the bottle did not originally have 360 pills. The manufacturer themself may have data on this. I’m not sure how much they care about the error of being plus or minus a few pills in a 360 pill bottle. I would guess that they fill the bottles by weight, which should be fairly accurate, but it’s unclear how accurate the bottler should aim to be. A quantity +/- 1 would certainly be unacceptable in, say, a dozen eggs, but it’s more conceivably okay in a 360 pill bottle. Who would notice, besides me?

Even if the manufacturer had the data on this based on their own internal random sampling, I’m not sure they’d share it, so I would probably have to buy a large sample of bottles and have them counted myself to make a determination as to whether incorrect quantities are a common occurrence. Though even if I were to do this, which would be very impractical, I’d still only have a probability that my bottle had the incorrect amount. I still might have made an error.

It’s much harder to think of how to quantify the odds that I made an error. There’s an observer of tracking bias if I were to try to log or track whether I do indeed take two pills each day. And then there’s the chance that even if I do accurately log consuming two pills each day that at some point I accidentally drop or lose one without realizing. The more I think about it, the more striking it is to realize just how hard it is to estimate whether I made some kind of mistake or if the bottle just had the wrong number of pills, post-facto.

If I really wanted to be sure of the cause of error, though, it would be simple to design a process to avoid this in the future. I would get multiple scales of very high precision and weight and manually count a few bottles to understand the relationship between weight and the number of pills in a bottle. If the weight is consistent, weight would be an excellent way to check each new bottle and make sure it has the correct number of pills. Given that baseline, I would know that if I end up with one pill at the end of the bottle, the error was most likely mine.

If the manufacturers are indeed prone to selling bottles with the wrong number of pills, this would also help keep them on tabs. As it is, lacking a such a system, they have plausible deniability, and that itself would allow mistakes to perpetuate. I doubt that anyone cares about this enough to make it worth doing this, but it is fun to consider the interplay between knowledge, certainty, incentives, and actions. And, costs notwithstanding, I would find it satisfying to know with certainty.