According to recent advances in particle theory, protons should decay 
with a half-life of at least 10^31 years.

(a) Particle physicists design an experiment to test this theory.  They take 
1000 cubic meters of super-pure water and watch for the by-products of proton
decay.  If all the protons in the water tank are equally likely to decay, and
the proton decay half-life is in fact 10^31 years, how many protons will decay
each year?

(b) After 7 years of operating this experiments, none of the protons in the
water tank have been observed to decay.  What is the shortest half-life that
a proton is likely to have, if this result is correct?