Fair value of a die with up to three rolls

Question

A casino lets you roll a fair six-sided die and pays you the face value. After the first roll you may stop or roll again; after the second you may stop or take one last, third roll, which you must accept. You are paid the value of whichever roll you stop on. What is the game worth with optimal play?

Short answer

$14/3$ . Work backwards. The last roll is worth $3.5$ , so on roll two keep $4,5,6$ : value $\tfrac12\cdot5+\tfrac12\cdot3.5=4.25$ . On roll one keep only $5,6$ : value $\tfrac13\cdot5.5+\tfrac23\cdot4.25=\tfrac{14}{3}$ .

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

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