Closest to a loaded die
A 12-sided die shows 12 with probability , and each of with probability (the remaining shared equally). You pick a number from 1 to 12, then your opponent picks a different number. The die is rolled, and whoever picked the number closer to the result wins; an exact tie in distance splits the win.
Both choose optimally. What is your probability of winning?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Picking 12 looks obvious. What does your opponent reply?
Your opponent will sit right next to you, on the side that captures more probability.
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s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Whatever you pick, your opponent sits just next to you, below or above, and takes everything on that side. So you want a number where both sides are as balanced as possible, keeping the heavy 12 on your side.
Step 1: the obvious pick fails. If you pick 12, your opponent picks 11 and wins every roll from 1 to 11: probability . You win only .
Step 2: pick 10. Your opponent's best replies:
- 9: they take 1 to 9, probability ; you keep 10, 11, 12: .
- 11: they take 11 and 12, ; you keep 1 to 10, .
Other replies are worse for them. So picking 10 guarantees .
Step 3: nothing better. With 11, the reply 10 leaves you only 11 and 12: . With 9 or lower, the reply one above takes the 12 and more than half of the probability. So the best you can do is .
The lesson. In games where the opponent moves second, choose the position that leaves them the least, not the one that is best on its own.