Dice Game 3
You're given a gamble with two dice: one 8-sided and one 12-sided. First, you roll the 8-sided die. After seeing its outcome, you can either keep that value or roll the 12-sided die (and must accept that result). What strategy maximizes your expected payout, and what is this expected payout?
Answer
The strategy is to compare the 8-sided roll to the expected value of the 12-sided die (which is 6.5).
- If the 8-sided die shows 7 or 8, keep it.
- If it shows 1 through 6, roll the 12-sided die.
Expected payout calculation:
\[
\begin{aligned}
\text{Probability of }1\text{–}6 &= \frac{6}{8}, \quad \text{with EV } = 6.5, \\
\text{Probability of }7\text{ or }8 &= \frac{2}{8}, \quad \text{with average value } = 7.5.
\end{aligned}
\]
Thus,
\[
\text{Expected Payout} = \frac{6}{8} \times 6.5 + \frac{2}{8} \times 7.5 = 4.875 + 1.875 = 6.75.
\]
The optimal strategy yields an expected payout of $6.75.
Notes and comments
Comment 1: Note that the optimal strategy here was easy and did not warrant much calculations.