$N$ independent and identically distributed random variables are sampled
sequentially from a known continuous distribution. The optimal policy for
maximizing the probability of obtaining the maximum of the sequence is
formulated generally for any information source and illustrated with several
examples. When following the optimal policy, it is shown in each case
considered that the expected number of random variables sampled is simply
related to the probability of selecting the maximum of the sequence. This
relationship is posed as a conjecture for all information sources.