Gambler's Ruin solved "which barrier is hit first, and how long does it take" for one specific chain (a random walk between two absorbing points) using first-step analysis. This lesson generalises both questions — exit distribution and exit time — to any finite Markov chain with one or more absorbing states, and proves the uniqueness theorems that justify the "guess and verify" method used informally in Gambler's Ruin. The generalisation matters because most real absorbing-chain questions (graduation vs. dropout, game outcomes, waiting times for a pattern) are not simple one-dimensional walks.
Learning objectives. By the end of this note you should be able to:
- set up the first-step equations for an exit probability and an exit time for a general finite chain;
- state and apply the uniqueness theorems for both;
- solve exit-distribution and exit-time problems using matrix inversion, ;
- use a stationary-distribution shortcut to compute certain waiting times without first-step analysis.
In this lesson
- Exit distributions
- Exit times
The full lesson, with worked examples and exercise solutions, is for members.
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