I came across an interesting question on the “/r/learnmath” subreddit. The poster describes two players, A and B, playing a game of dice. Each one has a die that has a specific side weighted. Each player rolls his die. If it lands on the weighted side, he gets the value of the side added to his score and rolls again. If it lands on any of the nonweighted sides, his turn is over. Whoever has the highest score wins. I found this question intriguing, and thought it would be a good opportunity for me to work on understanding infinite series, a weak area for me. Note that the original question states that if the first roll is a nonweighted side, the player gets that many points (instead of zero like on subsequent rolls). I’m going to ignore that rule and focus on the general pattern.
Problem: Each player has a die that is weighted on one side. He gets that many points and gets to roll again if it lands on the weighted side; if it lands on the unweighted side, his turn is over. More specifically, A’s weighted side is 3 and he has a 20% chance of rolling it; B’s weighted side is 2 and she has a 30% chance of rolling it. What is the chance that A will win (not tie)?