Suppose me and an opponent are racing to 100 points. We take turns rolling some die and adding the roll to our score. If the die is fair (and we randomly determine who goes first) it's clear this game is completely 50/50.
However, I'm interested in calculating the probability when the die isn't fair or when we each have different score targets.
For example, suppose we're playing with a fair 6-sided die but I only have to reach 90 points to win while my opponent must reach 100. Alternatively, suppose we're both racing to 100 but I have a 7-sided die and my opponent only has a 6-sided die.
While I'm able to determine these probabilities with great precision using Monte Carlo simulations I'm unsure how to derive these figures mathematically.
It seems promising to calculate the "average number of rolls required" for both players and compare these. Clearly, the player with the lower of these two values will be the favorite, however, I need some help in calculating the actual probability itself.