Compute gambler’s ruin probabilities and hitting times
Gambler's Ruin Probabilities and Hitting Times
A gambler starts with an initial wealth of i dollars. The game has two absorbing boundaries: ruin at 0 dollars and a target at N dollars.
In each round, the gambler's wealth changes as follows:
- With probability
p, wealth increases by 1. - With probability
q = 1 - p, wealth decreases by 1.
Assume all bets are independent.
- Derive the probability
u_ithat the gambler reaches the targetNbefore reaching ruin at 0, starting with wealthi. - Derive the expected number of steps until absorption (hitting either 0 or
N) starting from wealthi. - For the unbiased case where
p = q = 1/2, relate your results to a simple symmetric random walk hitting the boundaries {0, N}.
AI interview prep
Core dilemma
Balancing the mathematical derivation of probabilities and expected values with the understanding of underlying stochastic processes and their discrete-time nature.
2 tradeoffs and 2 stress tests locked
Interview prep
Unlock the interviewer X-Ray rubric, tradeoff matrix, stress tests, clarifying questions, and key concepts.
Compute gambler’s ruin probabilities and hitting times
Gambler's Ruin Probabilities and Hitting Times
A gambler starts with an initial wealth of i dollars. The game has two absorbing boundaries: ruin at 0 dollars and a target at N dollars.
In each round, the gambler's wealth changes as follows:
- With probability
p, wealth increases by 1. - With probability
q = 1 - p, wealth decreases by 1.
Assume all bets are independent.
- Derive the probability
u_ithat the gambler reaches the targetNbefore reaching ruin at 0, starting with wealthi. - Derive the expected number of steps until absorption (hitting either 0 or
N) starting from wealthi. - For the unbiased case where
p = q = 1/2, relate your results to a simple symmetric random walk hitting the boundaries {0, N}.