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Probability II / Random Walks on the Integers

Derivation — Solving the Gambler's Ruin Difference Equation

The previous note stated the closed-form absorption probability h(x)h(x) and expected duration D(x)D(x) for gambler's ruin without deriving them. This note carries that algebra in full — both because the technique (linear difference equations with constant coefficients, solved via a characteristic-root ansatz) is genuinely reusable, and because the non-fair case (p≠qp\ne q) is where the practically important asymmetry between winning and losing strategies becomes visible in closed form.
Learning objectives. By the end of this note you should be able to:
  • solve a linear second-order difference equation with constant coefficients via the characteristic-root method;
  • handle the degenerate case (repeated root) that arises exactly at p=q=12p=q=\tfrac12;
  • derive the expected-duration formula for both the fair and non-fair games.

In this lesson

  1. Solving for the absorption probability
  2. Solving for the expected duration

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