Two players until one player wins three games in a row. Each player will win with probability $\frac{1}2$. How many games will they play? 
QUESTION: Suppose two equally strong tennis players play against each other until one player wins three games in a row. The results of each game are independent, and each player will win with probability $\frac{1}2$. What is the expected value of the number of games they will play?


MY APPROACH: I have tried to set up some kind of recurrence relation here but could not succeed.. Observe that there can be at most a winning streak of $2$. A winning streak of $3$ means that the game ends.. If we assume that the number of $1$ game winning streak is $x$ and the number of $2$ game winning streak is $y$ then $x+y+1$ obviously yields the desired answer..
Note: A $1$ game winning streak simply means that they win alternately.. Since each one of them has a $50\%$ chance of winning therefore we can do this..
Now, somehow, we have to find the value of $x$ and $y$.. But here I am stuck.. With so less information, neither can I set up a recurrence relation, nor do I see any way to compute two variables..
Any help will be much appreciated.. :)
 A: Suppose the winner of the last game is on a $1$-game streak. How many more games until someone is on a $2$-game streak? This is just a geometric random variable with parameter $1/2$, so has expectation $2$.
Now, once someone is on a $2$-game streak, either they get a $3$-game streak next game, or you go back to someone being on a $1$-game streak. So from a $1$-game streak, after an expected $3$ games either someone completes a $3$-game streak or you are back where you started. The number of times this happens before you get a $3$-game streak is also exponential with parameter $1/2$, so has expectation $2$. Crucially, the number of stages of this form you have to go through is independent of the length of each stage. So the total time for all stages has expectation $2\times 3=6$. This is the expected time from the position where someone is on a streak of $1$, i.e. the expected number of games needed after the first game, so the total expectation is $7$.
A: Recurrence works fine.
All we care about is the length of the current win streak, we don't even care who has been winning.  Accordingly, let $E_i$ denote the expected number of games it will take if one player currently has a winning streak of length $i$.  The answer we seek is $E_0$.
We get: $$E_2=\frac 12\times 1+\frac 12\times (1+E_1)=1+\frac 12\times E_1$$
Similarly $$E_1=1+\frac 12\times (E_1+E_2)$$ and $$E_0=1+ E_1$$
This is easily solved and yields $$\boxed {E_0=7}$$
