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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 <var>S</var> = 0. 
The first player, Louise, adds to <var>S</var> different random numbers from the generator until <var>S</var> &gt; 1 and records her last random number '<var>x</var>'. 
The second player, Julie, continues adding to <var>S</var> different random numbers from the generator until <var>S</var> &gt; 2 and records her last random number '<var>y</var>'. 
The player with the highest number wins and the loser washes the dishes, i.e. if <var>y</var> &gt; <var>x</var> 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 <var>x</var> = 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 <var>y</var> = 0.91.
Since <var>y</var> &gt; <var>x</var>, 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

​x
 
<p>Julie proposes the following wager to her sister Louise.<br />She suggests they play a game of chance to determine who will wash the dishes.<br />For this game, they shall use a generator of independent random numbers uniformly distributed between 0 and 1.<br />The game starts with <var>S</var> = 0.<br />The first player, Louise, adds to <var>S</var> different random numbers from the generator until <var>S</var> &gt; 1 and records her last random number '<var>x</var>'.<br />The second player, Julie, continues adding to <var>S</var> different random numbers from the generator until <var>S</var> &gt; 2 and records her last random number '<var>y</var>'.<br />The player with the highest number wins and the loser washes the dishes, i.e. if <var>y</var> &gt; <var>x</var> the second player wins.</p>​<p>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 <var>x</var> = 0.44.<br />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 <var>y</var> = 0.91.Since <var>y</var> &gt; <var>x</var>, the second player wins.</p>​<p>Louise thinks about it for a second, and objects: "That's not fair".<br />What is the probability that the second player wins?<br />Give your answer rounded to 10 places behind the decimal point in the form 0.abcdefghij</p>

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