The pill problem can be generally stated as follows:

I have a bottle with N pills. Each day I need to take a dose of half a pill. When I remove a whole pill, I break it in half and take one half and put the other half back in the bottle. When a half pill is taken out, I swallow it as that day's dose. I shake the pills each day, so we can assume that each pill in the bottle is as likely to be taken out as any other. Further, I'm too lazy to only pull out half pills, so the first pill I touch is the one I pull out. Now when I have one dose left, we know that it is a half pill. But the day before, there were two doses. That means there could have been a whole pill or two half pills. What is the probability that there was a whole pill the day before my last dose?

Part 1: By hand, solve this for N=1, N=2, and N=3.

[Keep in mind that probability is the number of ways to succeed divided by the number of ways possible.We want to divide the number of ways to have a whole pill the day before the last dose by the number of ways we could have arrived at that day before the last dose.]