Research and trading rounds.
Gambler's ruin
The question
You start with 10 and bet 1 per round on a fair coin, stopping at 0 or at 20. What is the probability you reach 20, and how long does the game last on average?
Have a go before you open the solution. If you are stuck after a few minutes, the first paragraph below is usually enough of a nudge.
Show the worked solution
The answer
Probability 1/2, and expected duration 100 rounds.
Your wealth is a martingale, so its expectation at the stopping time equals its starting value: 20p = 10 gives p = 1/2.
For duration, note that wealth squared minus the number of steps is also a martingale.
That gives expected duration = a times (N - a) with a = 10 and N = 20, so 100 rounds.
The general fair-game results are p = a/N and duration a(N-a), and both are worth being able to produce without derivation.
What it is testing
Whether you reach for a martingale rather than a difference equation.
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