How long does the fair ruin game last?
You start with \40$1p = 0.5$100$0$.
On average, how many bets until the game ends (either boundary)? Derive the general fair-game formula.
Show a hint
Let be the expected number of bets from \iD_i = 1 + \tfrac12 D_{i+1} + \tfrac12 D_{i-1}D_0 = D_N = 0$. Try a quadratic.