Assume there are N pills

Probability of drawing a half pill "after" 0 trials = 0 (since all are whole pills)

Probability of drawing a half pill "after" 1 trials = 1/N (only one is half pill and others are whole pills)

Now, if half pill was drawn in the second trial,

Probability of drawing a half pill "after" 2 trials = 0

or, if a full pill was drawn in the second trial,

Probability of drawing a half pill "after" 2 trials = 2/N

So, In general,

Probability of drawing a half pill "after" 2 trials = 1/N*0 + (1-1/N)*2/N = 2(N-1)/N²

Continuing with this is somewhat difficult...

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Harish