A gambler with i chips plays a fair-or-loaded coin against the house, one chip per bet, until they either reach the target N or go broke: the two endings are absorbing states, so this deceptively small random walk is a full absorbing CTMC — a toy version of the extinction chain in the rare-events demo. Every quantity about it has an exact closed form: the ruin probability and the expected number of bets come from solving the same linear recurrence the rare-event machinery uses, one bet at a time.

The bars give the exact ruin probability from every possible starting bankroll; the highlighted one is your current start.