Let's try a short modified version of Risk where A starts with 4 armies and B starts with 2 armies. They both roll a 6-sided die at each turn: A will roll up to the number of dice equal to their number of armies minus 1 up to a maximum of 2 (so they will roll 1 or 2 dice when they have 3 armies, 1 if they have 2, and can not attack with 1 army); while B always rolls one. They will each continue to roll dice until either A has one army or B has none (in which case A wins). For a roll, if the maximum of A's rolls is bigger than B's, then B loses an army; if the max for A is lower, they lose an army; if the max for A is the same as the roll for B, no army is lost. For given values of m and n, set up the transition matrix for two cases: A always rolls the maximum number of dice; A always rolls one die. Find the probability that A wins and how long it takes in both cases.

Here is an example question and answer.