Consider the following dynamic card game with a regular deck of 26 red cards and 26 black cards. A dealer draws the unturned cards one by one, and we can ask him to stop at any time. For every red card drawn, we get 1 dollar and lose 1 dollar for every black card drawn. The problem consists in finding an algorithm which returns the expected value of the game. If we denote by $b$ and $r$, respectively, the number of black and red cards left in the deck at any time, the expected value of the game $E(b,r)$ satisfies:
with boundary conditions $E(0,r)=0$ and $E(b,0)=b$. The expected value of the game is therefore given by $E(26,26)$.
My question is, if we implement the recursive algorithm associated with the above formula, how can we determine its complexity? Using the trivial cases of $E(1,1)$ and $E(2,2)$, it would appear that we are dealing with exponential complexity, but is there a way to prove this properly, and if so, what is the number of necessary operations to compute $E(n,n)$ for an arbitrarily large integer $n$? Any ideas or references to literature would be greatly appreciated.