Unfair wager

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 > 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 example, 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

Written by gamwe6 who lives and works in San Francisco building useful things. You should follow him on Twitter