Keep the d8 or gamble on the d12?

EasyProbability Expected Value~4m

You roll a fair $8$ -sided die once. You may keep its value, or discard it and instead roll a fair $12$ -sided die once, taking that value. Under optimal play, what is your expected payout?

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The reroll is worth $\mathbb{E}[d_{12}]=6.5$ . Keep the $d_8$ only when it exceeds that.
Keep $\{7,8\}$ , else take $6.5$ : $\tfrac{6\cdot6.5+7+8}{8}$ .

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6.75 (± 0.01)

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Keep the d8 or gamble on the d12?

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