Throwing a Hundred Sided Dice

[Puzzle Prime]

Two people are taking turns throwing a hundred sided die. The game continues until someone throws a number larger than the number thrown by the opponent on the previous turn. What is the probability that the person who throws first wins the game?

Example: 4 – 3 – 3 – 2 – 5 => First player wins
Example: 6 – 3 – 2 – 1 – 1 – 4 => Second player wins