Eccoci.

Julie proposes the following wager to her sister Louise.
She suggests they play a game of chance to determine who will wash the dishes.
For this game, they shall use a generator of independent random numbers uniformly distributed between $0$ and $1$.
The game starts with $S = 0$.
The first player, Louise, adds to $S$ different random numbers from the generator until $S \gt 1$ and records her last random number '$x$'.
The second player, Julie, continues adding to $S$ different random numbers from the generator until $S \gt 2$ and records her last random number '$y$'.
The player with the highest number wins and the loser washes the dishes, i.e. if $y \gt x$ the second player wins.

For example, if the first player draws $0.62$ and $0.44$, the first player turn ends since $0.62+0.44 \gt 1$ and $x = 0.44$.
If the second players draws $0.1$, $0.27$ and $0.91$, the second player turn ends since $0.62+0.44+0.1+0.27+0.91 \gt 2$ and $y = 0.91$. Since $y \gt x$, the second player wins.

Louise thinks about it for a second, and objects: "That's not fair".
What is the probability that the second player wins?
Give your answer rounded to $10$ places behind the decimal point in the form 0.abcdefghij.