Now consider a modified version of Problem where the defense is also allowed to roll multiple dice. Each player's highest roll is compared with the other player's highest roll, their second highest roll is compared with the other player's second highest roll, etc. As before, any ties go to the defense.

(a) Suppose both players roll two dice. In this case, there are two armies to be lost since there are two dice comparisons (highest vs. highest and lowest vs. lowest). Find each of the following probabilities:

(i) Offense wins both comparisons (and thus defense loses two armies).

(ii) Offense wins one comparison and defense wins the other (and thus each lose one army).

(iii) Defense wins both comparisons (and thus offense loses two armies).

(b) Repeat all the calculations in part (a) for the scenario where the offense rolls three dice and the defense rolls two dice. As before, there are two comparisons to be made in this scenario (highest vs. highest and second highest vs. second highest).