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Let $w(a,b)$ be the expected profit with $a$ positive cards and $b$ negative cards. Then $w(0,0) = 0$ and $$ w(a,b) = \max\bigl(0, \tfrac{a}{a+b} (w(a-1,b) + 1) + \tfrac{b}{a+b} (w(a,b-1) - 1)\bigr). $$ Indeed, if we stop the game immediately, the profit is zero. Otherwise, with probability $\tfrac{a}{a+b}$, we pull out a positive card, and with probability $...

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