I'm solving some dice problems and came across this one.
You are invited to partake in a game with an 8-sided die. Here's how it works:
You get a chance to roll the die once, observe its outcome, and then decide if you'd like to stop or take a second roll. If you choose to stop, your score is the value from that single roll. If you choose to roll again, your score is the sum of both rolls. However, if the combined value surpasses 8, you "bust" and end up with no points.
What strategy would you adopt for this game? Specifically, after the outcome of the first roll, how would you decide whether to stop or roll again?
After thinking about it, intuitively I'd roll again if my first roll was 1, 2 or 3. I was ambiguous about 4 because chance of bust would be 50%.
The strategy I would adopt is this: if the expected value of rolling again after obtaining $k$ on the 1st roll is greater than $k$, then you should roll again. Otherwise don't. I think this is a reasonable answer as it maximises EV.
But then I got a surprising result. You should NOT roll again if your first roll was a 4. This is because
$\mathbb{E}[\text{score after rolling again}|\text{1st roll was }4]=\frac{4}{8}\cdot 0+\frac{1}{8}\cdot 5+\frac{1}{8}\cdot 6+\frac{1}{8}\cdot 7+\frac{1}{8}\cdot 8=3.25<4$.
Thus, only roll again if your first roll was $1,2$ or $3$.
Have I gone about this in the right way? Interested if anyone has a better method. Thanks!