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Compute gambler’s ruin probabilities and hitting times

ImcData Scientist · Onsite
MediumData ScientistOnsiteMachine Learning

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This question tests your ability to model a discrete random walk as a Markov chain, derive the probability of reaching an absorption state, and calculate the expected time to reach that state.

The Gambler's Ruin

A gambler starts with an initial capital of n dollars. In each round, they bet 1 dollar. With probability p, they win the dollar (capital becomes n+1), and with probability q = 1 - p, they lose the dollar (capital becomes n-1). The game ends if the gambler reaches a capital of 0 (ruin) or a target capital of N (success).

Task 1: Probability of Success

Derive the probability P(n) that the gambler reaches N before reaching 0, starting with capital n.

Task 2: Expected Hitting Time

Let E(n) be the expected number of rounds until the game ends (either reaching 0 or N), starting with capital n. Derive a formula for E(n) in the case where p = 0.5 (a fair game).

Examples

  • If n = 2, N = 4, and p = 0.5, the probability of hitting 4 before 0 is n/N = 2/4 = 0.5.
  • For the same parameters in a fair game, the expected time E(n) is n * (N - n) = 2 * (4 - 2) = 4 rounds.

Constraints

  • 0 <= n <= N
  • 0 < p < 1
  • Assume the process is memoryless and satisfies the Markov property.

Hints

Hint 1

For Task 1, set up a recurrence relation `P(n) = p * P(n+1) + q * P(n-1)` with boundary conditions `P(0) = 0` and `P(N) = 1`.

Hint 2

For Task 2, in a fair game, the recurrence is `E(n) = 1 + 0.5 * E(n+1) + 0.5 * E(n-1)`, which simplifies to a second-order difference equation.

Hint 3

Consider using the method of undetermined coefficients or characteristic equations to solve the homogeneous parts of the difference equations.

Follow-up questions

  • How does the probability of success `P(n)` change as `N` approaches infinity if `p > 0.5`?
  • If the gambler wins more than they lose (`p > 0.5`), how does the expected hitting time `E(n)` change compared to the fair game case?
  • How would you simulate this process using a Monte Carlo method to verify your theoretical results for large `N`?