Consider the following exercise from Lawler's Stochastic Calculus:
Suppose two people want to play a game in which person A has probability 2/3 of winning. However, the only thing that they have is a fair coin which they can flip as many times as they want. They wish to find a method that requires only a finite number of coin flips.
- Give one method to use the coins to simulate an experiment with probability 2/3 of success. The number of flips needed can be random, but it must be finite with probability one.
- Suppose $K < \infty$. Explain why there is no method such that with probability one we flip the coin at most K times.
Repeat the exercise with 2/3 replaced by $1/\pi$.
My guess is that the required game consists in the following stochastic process: we throw a coin for the $i$-th time, and if head happens we let $X_i=1$, otherwise $X_i=-1$.
Let $S_n=\sum_{i=1}^n X_i$. We start from $S_0=0$, and if $S_n=1$, we say that player A wins and the game halts, and if $S_n=-2$ we say that player B wins and we stop the game.
The number of flips needed to end the game can be proved (in greater generality) by letting $T_{a,b}=\min\{j : S_j=b \text{ or } S_j=-a\}$ be the stopping time, then by the optional sampling theorem we have $$ \begin{align} E(S_{T_{a,b}}) &= E(S_0)= 1\cdot\frac12 -1\cdot\frac12=0 \\ &= b\cdot P\{S_{T_{a,b}}=b\} - a\cdot P\{S_{T_{a,b}}=-a\}, \end{align} $$ which, using the fact that $P\{S_{T_{a,b}}=-a\}=1-P\{S_{T_{a,b}}=b\}$, gives $$ P\{S_{T_{a,b}}=b\} = \frac{a}{a+b}, $$ and in our case, $b=1$ and $a=2$, proves that player A indeed has winning probability of 2/3.
For the part two, namely proving that $$ P\{T_{a,b}<K\}=1, $$ I do not know how to proceed, and also I do not know how to modify the stochastic process so that the probability 2/3 is replaced by $1/\pi$.