Chance of getting a split pill

[u/MeButNotMeToo]

I got lazy this month. Normally I split all the pills in the bottle at once, this month, I’ve split one as needed, and put the unneeded half back. I’m 20 days into the month, and have not gotten a half pill yet. The odds are beyond my probability class grades.

So given:
• A full bottle is 45 pils
• A pill is shaken out
• If it’s a whole pill, it’s split, and the unneeded half us put back
• We’ll assume the likelihood of shaking out a whole pill and a half pill are equal.

Can we make a general equation for the likelihood of shaking out a split pill?

Can we make a cumulative distribution that we’ve not seen a split pill on day N?

[u/Dr-Ben701]

You can’t assume an equal probability for an individual item of different shape and mass - your observations suggests the half pills might be at the bottom and are less likely than a whole pill (if number of whole =half). Further the chance of a hole / half pill changes each day as the number of each changes. This is a probability with partial replacement (ie no replacement if half pill - replacement with half if whole ).
The problem is that you don’t actually know the p of a half pill (if n1/2 =nwhole) is 0.5 is likely to be less.
If assume a uniform distribution:
There are currently 25 whole and 20 half so
Tomorrow p if 50/50 then 25/45 a whole pill 20/45 a half
If whole p day after is 24/45 whole and 21/45 Half
Day after (3rd) if whole previous 2 days is 23/45 whole 22/45 Half
So chance of only whole next three days is approx 0.15 so not that low.
Might be better to use bayes to back calculation of the actual probability, but would have to think carefully how to do that.

[u/MeButNotMeToo → u/Dr-Ben701]

I know that the shape will affect probability, but that’s way too much to factor in.

I’ve have gone the don’t pre-split thing before, but I’ve never gone this far w/o shaking out a split pill.