Stochastic process problem, thank you!

2. (12 points) In a casino, you play a game of roulette wheel. Each round you cost $2 to spin the wheel. With equally chance, you will get either white or red result. You keep spinning the wheel until you get two reds of the three most recent spins and then you can win $9. What is your expected winning in this game? Hint: First solve the expected # of spins. Use the law of total expectation and separate it as two cases: (1) the first two spins both are red and you're done; (2) consider a Markov chain (Xm )0 Im_ (note the starting number of spins is n = 3) with state space S = [WWW. WWR. WRW. RWW. WRR. RWR]. Consider setting up appropriate absorbing states