Two players are competing in a game with a maximum of n points, where each turn only one player scores a point. If the first player has a points and the second b, what are the respective probabilities of winning, i.e. a=n or b=n, assuming both players have a 50% chance of winning each round?
Example Player 1: 4 points Player 2: 6 points Game won if 7 points are reached.
The probability that $A$ wins is a function $f$ of $n-a$ and $n-b$ only. Clearly, $f(0,k)=1$ for $k\ge 1$ and $f(k,0)=0$ for $k\ge 1$ ($f(0,0)$ is undefined or interpretable as $\frac12$), and we have the recursion $$\tag1f(x,y)=\frac{f(x-1,y)+f(x,y-1)}2 $$ for $x,y\ge1$. Quite obviously, we have $f(x,y)=1-f(y,x)$ and $f(x,x)=\frac12$. For small $x,y$, this should be enough to help you compute the desired result.
Example: $$f(7-4,7-6)=f(3,1)=\frac{f(2,1)+f(3,0)}2=\frac{\frac{f(1,1)+f(2,0)}2+f(3,0)}2 =\frac{\frac{\frac12+0}2+0}2=\frac18.$$ In fact, such an example with one player only one point away from a win is quite simple anyway: The other player needs to win several (here: three) times in a row ...