You throw two fair six-sided dice and are offered the product of the two numbers in dollars. Alternatively, you may pay $$4$ to throw both dice again exactly once, in which case you receive the new product (minus the fee). Playing rationally, what is your expected payout?

Keep a dice product or pay to reroll

Question

You throw two fair six-sided dice and are offered the product of the two numbers in dollars. Alternatively, you may pay $\text{\char36}4$ to throw both dice again exactly once, in which case you receive the new product (minus the fee). Playing rationally, what is your expected payout?

Short answer

$125/9 \approx \text{\char36}13.89$ . A fresh product averages $3.5^2=49/4$ , so a reroll is worth $49/4-4=33/4$ . Keep any product of $9$ or more and reroll on $8$ or less (probability $4/9$ ); averaging over the $36$ outcomes gives $\tfrac{125}{9}$ .

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

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