## Unfair wager

### Problem 436

Published on Saturday, 14th September 2013, 07:00 pm; Solved by 244Julie 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` > 1 and records her last random number '`x`'.

The second player, Julie, continues adding to `S` different random numbers from the generator until `S` > 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` > `x` the second player wins.

For e`x`ample, if the first player draws 0.62 and 0.44, the first player turn ends since 0.62+0.44 > 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 > 2 and `y` = 0.91.
Since `y` > `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