252.Twelve-Sided or Two Sixes
★★★optimal stoppingJane Street
You roll a fair 12-sided die and are paid the number shown. Instead of accepting it, you may discard the result and roll two fair six-sided dice, taking their sum as your payout. Playing optimally, what is the expected payout?
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Both options span 1 to 12 and 2 to 12, but they have very different spreads. Work out what the two-dice option is worth and switch only when the single die is below it.
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