MathDB

We consider the following game for one person. The aim of the player is to reach a fixed capital

C>2

. The player begins with capital

0<x_0<C

. In each turn let

x

be the player’s current capital. Define

s(x)

as follows:

s(x):=\begin{cases}x&\text{if }x<1\\C-x&\text{if }C-x<1\\1&\text{otherwise.}\end{cases}

Then a fair coin is tossed and the player’s capital either increases or decreases by

s(x)

, each with probability

\frac12

. Find the probability that in a finite number of turns the player wins by reaching the capital

C

Problem Statement