Assume I have an infinite bankroll and bet on black with an American roulette wheel , while using the "Martingale" strategy (doubling your betsize every time you lose). How can you go about calculating the expected profit at some number of rounds without generating every possible path? 1 dollar bet, 10 rounds possible paths + EV
As an example, you start with a bet size of 1 , given a probability of winning $P_{w} = 18/38$ and probability of loosing $P_{l} = 20/38$, using a martingale where every loss doubles bet size such that the bet size after $N$ losses is $2^{N}\$$ and the bet size after a win is equal to the starting bet.
What is the expected value after $N$ plays?
WW = 2 WL = 0 LW = 1 LL = -3 (18/38 * 18/38 * 2) + (18/38 * 20/38 * 0) + (20/38 * 18/38 * 1) + (20/38 * 20/38 * -3 ) = -.132 ^EV for 2 rounds of play by generating every path