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How can you simulate a fair coin with a biased coin? How can you simulate a die roll with a fair coin? How can you use a 10-sided die to simulate a 12-sided die?

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You have a coin with an unknown bias. How can you use it to produce a fair 50-50 outcome?You roll an 8-sided die and win the amount shown, or you may instead re-roll using a 12-sided die. How much should you pay to play?

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