Die with doubled even faces and one reroll

Question

A fair six-sided die pays its face value when it shows an odd number and double its face value when it shows an even number. After the first roll you may choose to roll once more, in which case the second result is the one that counts. Playing optimally, what is your expected payout?

Short answer

$7$ . A single roll pays $1,4,3,8,5,12$ for faces one to six, averaging $5.5$ , so you should keep only the payouts 8 and 12 (a four or a six) and reroll otherwise. The value is $\tfrac13\cdot10+\tfrac23\cdot5.5=7$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Counting and Combinatorics
Expectation, Mean, and Variance
Conditional Probability, Total Probability, and Bayes' Rule

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