You roll a fair six-sided die. You may either keep the value shown or re-roll exactly once, in which case you must accept the second roll. What is the expected payout under the optimal strategy?
Express as a fraction or decimal
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Re-roll if the first result is 1, 2, or 3 (since 3.5, the expected value of a re-roll, exceeds each). Keep if the result is 4, 5, or 6. The expected payout is 21⋅34+5+6+21⋅61+2+3+4+5+6=21(5)+21(3.5)=417.