What's the probability of losing a coin tossing gambling with a wealthy man? Imagine you have $k$ dollars in your pocket and you are gambling with a wealthy man (with infinitely much money). The rule is repeatedly tossing a coin and you win $\$1$ if it's a head, otherwise you lose $\$1$. Now, what's the probability that you get broke eventually?

The probability is $1$. This is the gambler’s ruin problem, and see also one-dimensional random walks.
