Higher Average Rolls Lower

The answer is $\boxed{\text{Yes}}$ . As an example using fair 6-sided dice, (fair meaning that each side has an equal chance of being on top), let dice $A$ have face numbers $1,1,1,1,1,8$ and dice $B$ have face numbers $2,2,2,2,2,2$ . Then the average roll value for dice $A$ is $\dfrac{1 + 1 + 1 + 1 + 1 + 8}{6} = \dfrac{13}{6}$ , while the average roll value for dice $B$ is $2$ . Yet of the $6 \times 6 = 36$ outcomes from rolling both dice simultaneously, for only $6$ of those outcomes will dice $A$ have a higher value than dice $B$ , i.e., dice $B$ will have a $5/6$ chance of having a higher individual roll.

An example where all side numbers are distinct is $1,2,3,4,5,37$ for dice $A$ and $6,7,8,9,10,11$ for dice $B$ .