Here is a version of the problem that got me into this field in earnest. In a growing population model — births λ·X, deaths μ·X — ask: what is the probability the population dies out by time T? With λ > μ this event is rare, and rare does not mean slow; it means invisible: a Monte Carlo simulation collecting a 10⁶-sample ensemble can easily record zero occurrences. The table below is computed on an absorbing chain: extinction converts probability mass to an “absorbed” ledger, and that ledger is what we sum.