Problem statement
You get to toss coins, each of which is either
of type H, having probabilities favouring heads
$\{(1\!-\!f),f\}$
or of type T, with probabilities
$\{f, (1\!-\!f)\}$, where $f$ is a known bias parameter such as $f=0.1$.
At each time step, one may either {\em{continue\/}} tossing the current coin,
or one may {\em{declare\/}} a guess for whether it is of type H or T.
Declarations are rewarded if accurate and penalized if incorrect.
The cost of continuing is zero.
The cost of declaring incorrectly is a penalty of $R_{-}$;
declaring correctly yields a reward of $R_{+}$.
To make the problem interesting, we set $R_{+} < R_{-}$.
The task is to choose a {\em{strategy}\/} that maximizes the
average return per unit time. We include no discounting in the problem.
Once you have declared for coin $t$ and been rewarded
or penalized, you get to move on to coin $n+1$. There is an infinite supply
of coins, all with the same bias $f$. The two types $H$ and $T$ are
equiprobable. Declaring takes no time: when you declare, you immediately
get to toss the next coin.
All postscript files are compressed with gzip - see this page for advice about gzip, if needed.