@mjd @gregeganSF Wow cool. To be pedantic, half pills left after you break the last whole pill and put half of it back in the container! Verified with a python script, below. I'll see if I can figure out an induction proof.

import fractions
F=fractions.Fraction
from collections import defaultdict

def update(d, rr):
e = defaultdict(F)
for (a,b),p in d.items():
if a == 1: rr[b+1] += p/(a+b)
if a>0: e[a-1,b+1] += p*a/(a+b)
if b>0: e[a,b-1] += p*b/(a+b)
return e, rr

def run():
rr = defaultdict(F)
d = {(30,0):F(1)}
for i in range(1,60):
d,rr = update(d, rr)
return d,rr

x,rr = run()
assert sum(rr.values()) == 1 # probabilities should sum to 1
e = sum(a*b for a,b in rr.items()) # expectation value
print(f'{e} =approx {float(e)}')
h30 = sum(F(1,i) for i in range(1,31)) # 30th harmonic number
assert e == h30