# What is the expected value of this game for large N?

For a given even $$N$$, I have $$N/2$$ red cards and $$N/2$$ black cards. Each time I draw a black card I win a dollar, each time I draw a red card I lose a dollar. I can stop at any time I like (and choose to do so in such a way that would maximize my expected winnings).

What is the expected value of the game for large $$N$$?

For a simple example, when $$N=2$$ - I would draw a card and if it's red, I would draw again (to get value of $$0$$), and if it's black I would stop, resulting in expected value of $$0.5$$.

To clarify, I know how to compute this numerically. I'm interested in the functional form for large $$N$$. For what it is worth, from simulations it appears to be $${\cal O}(\sqrt{N})$$.