Calculating probabilities over multiple dice rolls is easy, but what do you do if the amount of dice decreases (dependently) from roll to roll? This is a common feature of many games, including Risk, Axis & Allies, and the sort.
Example (from Axis & Allies):
Two teams have 2 and 3 dice, respectively. Each die on each team has a different value, called its 'strength', from 1 to 4. Team 1 has a 2 and a 3; Team 2 has a 1, a 1, and a 2. Each team rolls all of their dice. For every die that shows a number less than or equal to its strength (a 'hit'), the opposing team takes away their lowest-strength die. The teams then repeat until one team runs out of dice and loses.
How do you calculate the odds of one team winning over the course of the entire game? More importantly, can the solution be written as a formula, which could be used for any two teams of varying size and strength?
In general these sorts of problems rarely have a closed form solution, you rather have to brute force the entire matrix of probabilities. For larger numbers of dice this can get troublesome, but if the rules don't allow for easy analysis it's sort of what you are stuck with.
For the second part, "over the course of the entire game," is a bit tricky to understand, but the best way of doing that would be to build a decision tree and enumerate every possible move made and then determine the outcome. This very well may be outside of what a computer can do depending on how many moves are available at any given point in time, although if you can prune the list of moves down to "good moves," (i.e. alpha beta pruning), it's possible to get somewhere.
This is largely the premise of many artificial intelligences in various games such as chess or othello.