Gambler's ruin - Why is the probability of never reaching the barriers 0?

Question

I have found online this pdf which treats the gambler's ruin problem. However, in the first page the writer implicitly assumes that the gambler must reach one of the barriers (state $0$ or $N$) in finite time. It seems that it uses this implicit assumption to justify that the probability of ruin is $1-P_i(N)$. Is there a way to justify it without using that assumption? I think it has to do with that the probability of never reaching a barrier is $0$, but how can I prove it?

Answer 1

Note that if there were ever to be a sequence of $N$ winning bets in a row, the game must have ended at some point in that sequence (even if the gambler started at \$1, they would have ended up at \$N. If they started below \$1, the game ended at some earlier point). We can show the probability of ever getting a streak of $N$ wins is $1$.

Let $A_0$ be the event that we get a streak of $N$ heads starting from the $0$-th toss, $A_1$ be the event that we get a streak of $N$ heads starting from the $N+1$-st toss, and so on to define $A_n$. Note that $P(A_n) = p^N > 0$, and each $A_n$ is independent. The probability of getting a streak of $N$ heads is at least \begin{align*} \mathbb{P}(\bigcup_{n=1}^\infty A_n) = 1-\mathbb{P}(\bigcap_{n=1}^\infty A_n^c) = 1-\prod_{n=1}^\infty \mathbb{P}(A_n^c) = 1-\prod_{n=1}^\infty (1-q)^N = 1, \end{align*} so we conclude that the probability that the gambler's wealth hits a barrier at some point is $1$.

Answer 2

The game can only end in one of two ways - the gambler reaches their target, or goes broke. There is no other option besides win or lose - to say there is a non-zero probability of not reaching either barrier is to say that there is some fraction of games that end without winning or losing, but that defies the very premise of the game. "The game isn't over yet" is not a valid outcome of the game, the game is defined such that you must continue playing until you hit one of the barriers.