probability - Keep the value of an 8-sided die roll, or gamble by taking a 12-sided die roll. What's the best strategy?

Consider a dice game in which you try to obtain the largest number (e.g. you win a prize based on the final dice roll).

1. You roll an 8-sided die, with numbers 1–8 on the sides.


2. You may either keep the value you rolled, or choose to roll a 12-sided die, with numbers 1–12 on the sides.

What's the best strategy for choosing what to do in step #2?

I know the 8-sided die has expected payoff of 4.5, and the 12-sided die has expected payoff of 6.5. So I think relying on the 12-sided die is better — but how do I show the probability of this?

Answer

If you are trying to maximise the expected score, then since the expected value of the 12-sided die is $6.5$, it makes sense to stop when the 8-sided die shows greater than $6.5$, i.e. when it shows $7$ or $8$, each with probability $\frac18$. So with probability $\frac34$ you throw the 12-sided die.

The expected score is then $$7 \times \frac18 + 8 \times \frac18 + 6.5 \times \frac34 = 6.75.$$