A contest consists of rounds. In each round, each robot has a single attempt to score. An attempt works like this:
- Advancing. The running track is the number line. The robot starts at 0, the starting line, and the takeoff point is at 1. It moves by drawing a real number uniformly from [0, 1] and adding it to its current position. Every draw is independent of all the others.
- Deciding. After each advance, including the very first, the robot sees its new position and chooses to jump or to wait. If it waits, it advances again.
- Busting. If the robot crosses the takeoff point (its position reaches 1 or more) before jumping, the attempt scores 0.
- Jumping. If the robot jumps from a position x < 1, it draws one final real number u uniformly from [0, 1]. The attempt scores x + u.
In a head-to-head contest, the two robots each make one attempt per round, without knowing the other's result. If they tie (typically because both scored 0), that round is discarded and a new round begins. As soon as one robot scores higher than the other in the same round, that robot is declared the winner.
Both robots are programmed to maximise their probability of winning the contest, and each knows the other's strategy. So they play a symmetric equilibrium: both use the same strategy, and neither could raise its winning probability by switching to a different strategy.
You sit down to watch a match's very first attempt (in the first round, which may or may not end up being discarded).
Question. What is the probability that this attempt scores 0?
Give this probability as a decimal rounded to 9 digits after the decimal point.