You and an opponent are playing tennis - first to get $2$ wins in a row wins. The probability of you getting a win is $0.6$. The probability of him getting a win is $0.4$. What's the probability of you winning the game?
I think this can be modeled as a Markov chain with 5 states (2 Losses, 1 Loss, 0 net, 1 Win, 2 Wins). Therefore, I think I could write out some equations to solve this. Can someone tell me if this makes sense/if it's wrong?
P(you win right off the bat) $= (0.6)(0.6) = 0.36$
P(he wins right off the bat)$ = (0.4)(0.4) = 0.16$
P(you win)$ = \frac{0.36}{0.36+0.16}$
Answer:
Case 1: You win two games consecutively$ = 0.36$
Case 2: You win a game and your opponent loses a game$ = 0.24$
Case 3: You lsoe a game and your opponent wins a game$ = 0.24$
Case 4: You lose two consecutive games and your opponent wins $ = 0.16$
In both cases 2 and 3, the game can be viewed as draw and back to square one. Thus the probability that is not a winner is sum of case 2 and 3 $= 0.48$
The probability that you will win $= 0.36 + 0.48*(.36)+0.48^2*(.36) + \cdots \infty$
$= 0.36\frac{1}{(1-0.48)} = \frac{9}{13}$
The probability that your opponent will win $=0.16 + 0.48*(.16)+0.48^2*(.16) + \cdots \infty$
$= 0.16\frac{1}{(1-.48)} = \frac{4}{13}$
This is one way you can simplify the game and find the solution unless you know the Markov Chain way of solving.