Gambler's Ruin 10/21

Suppose we do a sequence of Bernouill(p) trials, we call them coin tosses, with a p-coin. A wins the toss if the coin comes up heads, and B wins if tails. If A wins he takes 1 from B, if not the inverse happens. The overall game is played until one of the players goes bankrupt.

Call $E_i=\{ A \; wins\; the\; whole\; game\; starting\; with\; i\; \$\}$

Then we do a ``one-step'' analysis, take P_i=P(E_i), we can show that:

$$\begin{eqnarray*}P_i &= &P(E_i\vert A\mbox{ wins first toss})P(A\mbox{ wins firs... ... & \left( ( \frac{q}{p})^{N-1}+\cdots +\frac{q}{p} \right)P_1\\ \end{eqnarray*}$$

Now we use the two boundary conditions: $P_0=0 \qquad P_N=1$, and the recurrence above leads to the following conclusions (using the sum of the geometric series of term $\frac{q}{p}$.

$$P_i=\left\{ \begin{array}{ll} \frac{1- ( \frac{q}{p})^i}{(1-\... ..._1= \frac{i}{N} &if \; \; \frac{q}{p} =1\\ \end{array}\right.$$