Fast Baseball Pitcher

For ease of computation, let's calculate the ELI, a stat I'm defining as ERA divided by 9. We want this to be equal than $\frac{1}{9}$ obviously. Let $f(x,y)$ designate the ELI with $x$ runners on and $y$ outs. Obviously, with $y=3$ our function resolves to zero. Further, with $y \neq 3$ $f(3,y) = pf(3,y+1) + (1-p)(1+f(3,y))$ . Even further, for $x, y \neq 3$ , $f(x,y) = pf(x,y+1) + (1-p)f(x+1,y)$ . What to do now? Well, we have a system of 16 equations and 16 unknowns, and we could solve $f(0,0)$ in terms of $p$ and set that equal to $\frac{1}{9}$ if we wanted. I opted to use Excel to get a numerical value for $f(0,0)$ in terms of a given $p$ , then using a digit-by-digit approximation to find the proper value.