Die roll payout with one optional reroll

Question

A casino pays you the face value of a fair six-sided die. After your first roll you may either take that amount or roll once more, in which case you must accept the second value. What is the fair price of this game?

Short answer

$4.25$ . A fresh roll is worth $3.5$ , so keep $4$ , $5$ or $6$ (average $5$ , probability $\tfrac12$ ) and reroll otherwise. The value is $\tfrac12\cdot5+\tfrac12\cdot3.5=4.25$ .

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

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Lessons behind this question

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

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