Question 85
Consider a three-person game involving the following three types of fair six-sided dice.
- Dice of type $A$ have faces labelled $2,2,4,4,9,9$.
- Dice of type $B$ have faces labelled $1,1,6,6,8,8$.
- Dice of type $C$ have faces labelled $3,3,5,5,7,7$.
All three players simultaneously choose a die (more than one person can choose the same type of die, and the players don't know one another's choices) and roll it. Then the score of a player $P$ is the number of players whose roll is less than $P$'s roll (and hence is either $0$, $1$, or $2$). Assuming all three players play optimally, what is the expected score of a particular player?
- Dice of type $A$ have faces labelled $2,2,4,4,9,9$.
- Dice of type $B$ have faces labelled $1,1,6,6,8,8$.
- Dice of type $C$ have faces labelled $3,3,5,5,7,7$.
All three players simultaneously choose a die (more than one person can choose the same type of die, and the players don't know one another's choices) and roll it. Then the score of a player $P$ is the number of players whose roll is less than $P$'s roll (and hence is either $0$, $1$, or $2$). Assuming all three players play optimally, what is the expected score of a particular player?