Now you may bet either color, can you beat 50%?

Question

A standard shuffled deck (26 red, 26 black) is dealt face-up one card at a time. When you stop, you may now bet the next card is red or black, your choice, using the counts you have seen. You may also run to the end. Can you now beat a 50\% win probability? What is the best strategy?

Accepted Answer

Answer: Yes, easily, up to a guaranteed win. The original problem is stuck at exactly 	frac12 only because you must pre-commit to red; freeing the color breaks the martingale. Why the constraint mattered. With a fixed target color, the probability the next card is that color, the red fraction remaining, is a martingale starting at 	frac12, so optional stopping pins every strategy at 	frac12. Once you may choose the color at the moment you stop, you are no longer betting a fixed martingale; you bet whichever side the current count favors. Strategy 1: bet the majority remaining color. If r r…

Now you may bet either color, can you beat 50%?

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