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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 ${\cal O}(\sqrt{N})$.

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