It’s coming to the end of the year, which can only mean one thing: time for this year’s Robot Javelin finals! Whoa wait, you’ve never heard of Robot Javelin? Well then! Allow me to explain the rules:

• It’s head-to-head. Each of two robots makes their first throw, whose distance is a real number drawn uniformly from [0, 1].
• Then, without knowledge of their competitor’s result, each robot decides whether to keep their current distance or erase it and go for a second throw, whose distance they must keep (it is also drawn uniformly from [0, 1]).
• The robot with the larger final distance wins.

This year’s finals pits your robot, Java-lin, against the challenger, Spears Robot. Now, robots have been competing honorably for years and have settled into the Nash equilibrium for this game. However, you have just learned that Spears Robot has found and exploited a leak in the protocol of the game. They can receive a single bit of information telling them whether their opponent’s first throw (distance) was above or below some threshold d of their choosing before deciding whether to go for a second throw. Spears has presumably chosen d to maximize their chance of winning — no wonder they made it to the finals!

Spears Robot isn’t aware that you’ve learned this fact; they are assuming Java-lin is using the Nash equilibrium. If you were to adjust Java-lin’s strategy to maximize its odds of winning given this, what would be Java-lin’s updated probability of winning? Please give the answer in exact terms, or as a decimal rounded to 10 places.