I am looking to find the probability of winning a basic dice roll-off using a 6 sided die if one of the players can re-roll their die. The main thing that is throwing me off is that player 2 can re-roll the die but doesn't have to, and if the first roll or the re-roll equals player 1's roll then the process restarts.
Example 1:
Player 1 - Rolls a 2
Player 2 - Rolls a 4 (win)
Example 2:
Player 1 - Rolls a 2
Player 2 - Rolls a 1, re-rolls and gets a 5 (win)
Example 3:
Player 1 - Rolls a 5
Player 2 - Rolls a 5
At this point player 2 can call it a tie and start fresh, or use his re-roll to attempt and roll a 6, although this doesn't seem to be his best option to win.
Example 4:
Player 1 - Rolls a 4
Player 2 - Rolls a 4
At this point player 2 decides to call it a tie, and then they both re-roll. Player 2 still has the ability to then re-roll his result in this round.
Player 1 - Rolls a 3
Player 2 - Rolls a 2, re-rolls and gets a 1 (loss)
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