What are the odds of never losing in a loaded coin game?

Question

Let's consider this simple dice game: A coin is faked so it has p chance to land on heads, and 1-p chance to land on tails. Every round costs $1, and gives you $2 if you win (for a total of +$1).

Assume you're starting with $n. What are your odds to "go infinite" - be able to play the game forever? This sounds like Markov Chains 101, it's just been ages since I read anything about Markov Chains.

Also - given any constant m, what are the odds of ever reaching $m in this game?

Answer 1

Define $f(n)$ as the probability of playing forever when starting out with n coins. Also, assume that the probability p of winning a round is bigger than $\frac{1}{2}$ (otherwise, the probability of playing forever is 0). Then, we get the recurrence relation $$ f(n) = p f(n + 1) + (1-p) f(n-1) $$ with the boundary conditions $f(0) = 0$ and $\lim_{n \to \infty} f(n) = 1$.

The general solution of the recurrence relation is $$ f(n) = a + b \left( \frac{1-p}{p} \right)^n $$ and from the boundary conditions we get $a = -b = 1$. So, the probability to play infinitely is $$ f(n) = 1 - \left( \frac{1-p}{p} \right)^n . $$

The second question can be answered just as the first one, but with different boundary conditions: $f(0) = 0$ and $f(m) = 1$. This leads to the probability of reaching m as $$ \frac{1 - \left( \frac{1-p}{p} \right)^n}{1 - \left( \frac{1-p}{p} \right)^m} . $$

By the way, the keyword for this kind of problems is 'Random Walk'.