## The Dice Game

Hello everyone! I was going through my sister's math practice sets and I found a probability question that is very challenging for me. I am trying to find a solution, but it is proving to be very difficult. Hopefully, someone can clear my concepts here. The question is as follows:

1. Consider a game with two players, Jim and Annie. Annie has a red die and Jim has a white die. They roll their dice and note the number on the upper face. Annie wins if her score is higher than Jim's (note that Jim wins if the scores are the same). If both players roll their dice once each what is the probability that Annie will win the game? (easy)

2. Now consider the same game where Annie can roll her die a second time and will note the higher score of the two rolls but Jim rolls only once. In this case, what is the probability that Annie will win? (easy, again )

3. In the case when both players can roll their dice twice, what is the Probability that Annie wins? (easy)

4. Investigate the game when they can roll the dice more than twice, but not necessarily the same number of times. (difficult )

I worked out the answers to 1. 2. and 3. as 15/36, 125/216 and 505/1296 respectively. However, part four has me stumped. Can anyone please point me in the right direction with this?