Probability · Maths EE idea · Solid
The gambler's ruin problem
A research question to start from
What is the probability that a gambler with £k reaches £N before going broke in a game with win probability p, and how does a small house edge change it?
A starting point, not your question: change the case, the comparison or the limit until it is yours. The research-question builder helps you check it.
Why it works as a maths EE
An exact answer from a recurrence, with a striking conclusion to discuss.
Mathematics you would need
- Random walks
- Second-order recurrence relations
- Conditional probability
- Expected duration
Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.
One possible line of attack
- Set up and solve the recurrence for the ruin probability.
- Treat the fair and unfair cases separately.
- Check with simulation and compute expected game length.
Scope and difficulty
Solid. Solid; expected duration is the stretch.
Pitfalls
- Simulation only.
- Dividing by zero in the fair case.
Where to start reading
Search a library catalogue or a university's open lecture notes for: gambler's ruin recurrence solution; expected duration random walk. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).