I'm part way through reading Mathematics of Poker and I'm currently on the first chapter of Part IV (on risk of ruin). It states that for games with positive expected value the risk of ruin is < 1.

Now this confused me when I thought about a theorem on absorbing states of Markov Chains I learnt last semester at University. I couldn't find my notes but I found the theorem on some other uni's notes. It's on page 9 of these notes https://math.dartmouth.edu/archive/m20x06/public_html/Lecture14.pdf. It states that for all markov chains where the probability of being absorbed is > 0 for all states, the probability the process will be absorbed is 1.

These two statements obviously contradict each other. So I assume there's something obvious that I'm missing, but I fail to see the difference between the problem of bankroll management and markov chains. I would have thought that our problem of bankroll management is just a markov chain with an absorbing state at 0.

Thanks for anyone who can help clarifying what I'm not understanding