\kproblem\ksource{David R's Brain} There is a casino in St. Petersburg
where you have the choice of betting any portion of your wealth on
either of two games: in the first game, you either receive 3 times
what you bet or lose it all with equal probability.  In the second
game the casino gives you 1.4 times what you bet with probability 1
(this casino is understandably very popular).  Each trial of the two
games are played simultaneously.

\bparts

\ppart Call $S_1$ and $S_2$ the fraction of your wealth you bet on the
two games (notice $S_1+S_2\leq 1$).  What values of $S_1$ and $S_2$
maximize the expected value of your wealth after 1 trial?  What is the
maximum expected value of your wealth?
\solution{$S_1=1$, max is 1.5}

\ppart What strategy maximizes your expected wealth after $n$ trials?
\solution{Always have $S_1=1$}

\ppart Using that strategy, what is the probability that after 10 trials, you have more money than you started with?
\solution{$1-2^{-10}$}

\ppart After $2n$ trials, what is the {\em most likely} number of
trials in game one that are wins.  And what is the probability that
this occurs?
\solution{$n$, $f_{2n,0.5}(n)\approx $}

\ppart What fraction of your wealth should you bet on each trial of
the two games, if you want to maximize your wealth after $2n$ trials
in the most likely case that you found above? (i.e. suppose that there
are that number of wins over the course of the $2n$ trials).  Assume
that it is in your interest to have $S_1+S_2=1$.
\solution{Maximize $(1.4+1.6S_1)^{0.5}(1.4-1.4S_1)^{0.5}$ over $S_1$.  $S_1=0.0625$.}
\eparts










